L^{2}-Invariants from the Algebraic Point of View

Wolfgang Lück^{∗} Fachbereich Mathematik Universität Münster Einsteinstr. 62 48149 Münster Germany

December 14, 2004

Abstract

We give a survey on L^{2}-invariants such as L^{2}-Betti numbers and L^{2}- torsion taking an algebraic point of view. We discuss their basic definitions, properties and applications to problems arising in topology, geometry, group theory and K-theory.

Key words: dimensions theory over finite von Neumann algebras, L^{2}-Betti numbers, Novikov Shubin invariants, L^{2}-torsion, Atiyah Conjecture, Singer Conjecture, algebraic K-theory, geometric group theory, measure theory. Mathematics Subject Classification 2000: 57S99, 46L99, 18G15, 19A99, 19B99, 20C07, 20F25.

0 Introduction

The purpose of this survey article is to present an algebraic approach to L^{2}- invariants such as L^{2}-Betti numbers and L^{2}-torsion. Originally these were defined analytically in terms of heat kernels. After it was discovered that they have simplicial and homological algebraic counterparts, there have been many applications to various problems in topology, geometry, group theory and algebraic K-theory, which on the first glance do not involve any L^{2}-notions. Therefore it seems to be useful to give a quick and friendly introduction to these notions in particular for mathematicians who have more algebraic than analytic background. This does not at all mean that the analytic aspects are less important, but for certain applications it is not necessary to know the analytic approach and it is possible and easier to focus on the algebraic aspects. Moreover, questions about L^{2}-invariants of heat kernels such as the Atiyah Conjecture or the zero-in-the-spectrum-Conjecture turn out to be strongly related to algebraic questions about modules over group rings.

∗email: lueck@math.uni-muenster.de www: http://www.math.uni-muenster.de/u/lueck/ FAX: 49 251 8338370

The hope of the author is that more people take notice of L^{2}-invariants and L^{2}-methods, and may be able to apply them to their favourite problems, which not necessarily come a priori from an L^{2}-setting. Typical examples of such instances will be discussed in this survey article. There are many open questions and conjectures which have the potential to stimulate further activities.

The author has tried to write this article in a way which makes it possible to quickly pick out specific topics of interest and read them locally without having to study too much of the previous text.

These notes are based on a series of lectures which were presented by the author at the LMS Durham Symposium on Geometry and Cohomology in Group Theory in July 2003. The author wants to thank the organizers Martin Bridson, Peter Kropholler and Ian Leary and the London Mathematical Society for this wonderful symposium and Michael Weiermann for proof reading the manuscript.

In the sequel ring will always mean associative ring with unit and R-module will mean left R-module unless explicitly stated differently. The letter G denotes a discrete group. Actions of G on spaces are always from the left.

Contents

0 Introduction 1

1 Group von Neumann Algebras 4 1.1 The Definition of the Group von Neumann Algebra . . . . . . . . 4 1.2 Ring Theoretic Properties of the Group von Neumann Algebra . 6 1.3 Dimension Theory over the Group von Neumann Algebra …. 7

2 Definition and Basic Properties of L^{2}-Betti Numbers 12 2.1 The Definition of L^{2}-Betti Numbers . . . . . . . . . . . . . . . . 12 2.2 Basic Properties of L^{2}-Betti Numbers . . . . . . . . . . . . . . . 14 2.3 Comparison with Other Definitions . . . . . . . . . . . . . . . . . 18 2.4 L^{2}-Euler Characteristic . . . . . . . . . . . . . . . . . . . . . . . 19

3 Computations of L^{2}-Betti Numbers 22 3.1 Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Finite Coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.3 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.4 Three-Dimensional Manifolds . . . . . . . . . . . . . . . . . . . . 23 3.5 Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.6 Spaces with S^{1}-Action . . . . . . . . . . . . . . . . . . . . . . . . 25 3.7 Mapping Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.8 Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4 The Atiyah Conjecture 29

4.1 Reformulations of the Atiyah Conjecture . . . . . . . . . . . . . . 29 4.2 The Ring Theoretic Version of the Atiyah Conjecture . . . . . . . 30 4.3 The Atiyah Conjecture for Torsion-Free Groups . . . . . . . . . . 32 4.4 The Atiyah Conjecture Implies the Kaplanski Conjecture …. 33 4.5 The Status of the Atiyah Conjecture . . . . . . . . . . . . . . . . 34 4.6 Groups Without Bound on the Order of Its Finite Subgroups . . 35

5 Flatness Properties of the Group von Neumann Algebra 36

6 Applications to Group Theory 37 6.1 L^{2}-Betti Numbers of Groups . . . . . . . . . . . . . . . . . . . . 38 6.2 Vanishing of L^{2}-Betti Numbers of Groups . . . . . . . . . . . . . 40 6.3 L^{2}-Betti Numbers of Some Specific Groups . . . . . . . . . . . . 41 6.4 Deficiency and L^{2}-Betti Numbers of Groups . . . . . . . . . . . . 43

7 G- and K-Theory 46 7.1 The K_{0}-group of a Group von Neumann Algebra . . . . . . . . . 46 7.2 The K_{1}-group and the L-groups of a Group von Neumann Algebra 49 7.3 Applications to G-theory of Group Rings . . . . . . . . . . . . . 49 7.4 Applications to the Whitehead Group . . . . . . . . . . . . . . . 51

8 L^{2}-Betti Numbers and Measurable Group Theory 52 8.1 Measure Equivalence and Quasi-Isometry . . . . . . . . . . . . . 53 8.2 Discrete Measured Groupoids . . . . . . . . . . . . . . . . . . . . 54 8.3 Groupoid Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 8.4 L^{2}-Betti Numbers of Standard Actions . . . . . . . . . . . . . . . 57 8.5 Invariance of L^{2}-Betti Numbers under Orbit Equivalence . . . . . 58

9 The Singer Conjecture 59 9.1 The Singer Conjecture and the Hopf Conjecture . . . . . . . . . . 59 9.2 Pinching Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 60 9.3 The Singer Conjecture and Kähler Manifolds . . . . . . . . . . . 61

10 The Approximation Conjecture 62

11 L^{2}-Torsion 64 11.1 The Fuglede-Kadison Determinant . . . . . . . . . . . . . . . . . 64 11.2 The Determinant Conjecture . . . . . . . . . . . . . . . . . . . . 67 11.3 Definition and Basic Properties of L^{2}-Torsion . . . . . . . . . . . 68 11.4 Computations of L^{2}-Torsion . . . . . . . . . . . . . . . . . . . . . 71 11.5 Some Open Conjectures about L^{2}-Torsion . . . . . . . . . . . . . 72 11.6 L^{2}-Torsion of Group Automorphisms . . . . . . . . . . . . . . . . 73 12 Novikov-Shubin Invariants 74

12.1 Definition of Novikov-Shubin Invariants . . . . . . . . . . . . . . 75 12.2 Basic Properties of Novikov-Shubin Invariants . . . . . . . . . . . 76 12.3 Computations of Novikov-Shubin Invariants . . . . . . . . . . . . 77 12.4 Open Conjectures about Novikov-Shubin invariants . . . . . . . . 78

13 A Combinatorial Approach to L^{2}-Invariants 80

14 Miscellaneous 82

References 82

Notation 91

Index 92

1 Group von Neumann Algebras

The integral group ring ℤG plays an important role in topology and geometry, since for a G-space its singular chain complex or for a G-CW -complex its cellular chain complex are ℤG-chain complexes. However, this ring is rather complicated and does not have some of the useful properties which other rings such as fields or semisimple rings have. Therefore it is very hard to analyse modules over ℤG. Often in algebra one studies a complicated ring by investigating certain localizations or completions of it which do have nice properties. They still contain and focus on useful information about the original ring, which now becomes accessible. Examples are the quotient field of an integral domain, the p-adic completion of the integers or the algebraic closure of a field. In this section we present a kind of completion of the complex group ring ℂG given by the group von Neumann algebra and discuss its ring theoretic properties.

1.1 The Definition of the Group von Neumann Algebra

Denote by l^{2}(G) the Hilbert space l^{2}(G) consisting of formal sums ∑_{g∈G} λ_{g} · g for complex numbers λ_{g} such that ∑_{g∈G} |λ_{g}|^{2} < ∞. The scalar product is defined by

⟨∑_{g∈G} λ_{g} · g, ∑_{g∈G} μ_{g} · g⟩ := ∑_{g∈G} λ_{g} · \overline{μ_{g}}.

This is the same as the Hilbert space completion of the complex group ring ℂG with respect to the pre-Hilbert space structure for which G is an orthonormal basis. Notice that left multiplication with elements in G induces an isometric G-action on l^{2}(G). Given a Hilbert space H, denote by B(H) the C^{∗}-algebra of bounded (linear) operators from H to itself, where the norm is the operator norm and the involution is given by taking adjoints.

Definition 1.1 (Group von Neumann algebra). The group von Neumann algebra N (G) of the group G is defined as the algebra of G-equivariant bounded operators from l^{2}(G) to l^{2}(G)

N (G) := B(l^{2}(G))^{G}.

In the sequel we will view the complex group ring ℂG as a subring of N (G) by the embedding of ℂ-algebras ρ_{r}: ℂG → N (G) which sends g ∈ G to the G-equivariant operator r_{g^{−1}} : l^{2}(G) → l^{2}(G) given by right multiplication with g^{−1}.

Remark 1.2 (The general definition of von Neumann algebras). In general a von Neumann algebra A is a sub-∗-algebra of B(H) for some Hilbert space H, which is closed in the weak topology and contains id: H → H. Often in the literature the group von Neumann algebra N (G) is defined as the closure in the weak topology of the complex group ring ℂG considered as ∗-subalgebra of B(l^{2}(G)). This definition and Definition 1.1 agree (see [60, Theorem 6.7.2 on page 434]).

Example 1.3 (The von Neumann algebra of a finite group). If G is finite, then nothing happens, namely ℂG = l^{2}(G) = N (G).

Example 1.4 (The von Neumann algebra of ℤ^{n}). In general there is no concrete model for N (G). However, for G = ℤ^{n}, there is the following illuminating model for the group von Neumann algebra N (ℤ^{n}). Let L^{2}(T^{n}) be the Hilbert space of equivalence classes of L^{2}-integrable complex-valued functions on the n-dimensional torus T^{n}, where two such functions are called equivalent if they differ only on a subset of measure zero. Define the ring L^{∞}(T^{n}) by equivalence classes of essentially bounded measurable functions f: T^{n} → ℂ, where essentially bounded means that there is a constant C > 0 such that the set {x ∈ T^{n} | |f(x)| ≥ C} has measure zero. An element (k_{1}, …, k_{n}) in ℤ^{n} acts isometrically on L^{2}(T^{n}) by pointwise multiplication with the function T^{n} → ℂ, which maps (z_{1}, z_{2}, …, z_{n}) to z_{1}^{k_{1}} ·. . .·z_{n}^{k_{n}}. Fourier transform yields an isometric ℤ^{n}-equivariant isomorphism l^{2}(ℤ^{n}) ⟶^{≅} L^{2}(T^{n}). Hence N (ℤ^{n}) = B(L^{2}(T^{n}))^{ℤ^{n}}. We obtain an isomorphism (of C^{∗}-algebras)

L^{∞}(T^{n}) ⟶^{≅} N (ℤ^{n})

by sending f ∈ L^{∞}(T^{n}) to the ℤ^{n}-equivariant operator

M_{f}: L^{2}(T^{n}) → L^{2}(T^{n}), g ↦ g · f,

where g · f(x) is defined by g(x) · f(x).

Let i: H → G be an injective group homomorphism. It induces a ring homomorphism ℂi: ℂH → ℂG, which extends to a ring homomorphism

N (i): N (H) → N (G)  (1.5)

as follows. Let g: l^{2}(H) → l^{2}(H) be a H-equivariant bounded operator. Then ℂG ⊗_{ℂH} l^{2}(H) ⊆ l^{2}(G) is a dense G-invariant subspace and

id_{ℂG} ⊗_{ℂH}g: ℂG ⊗_{ℂH} l^{2}(H) → ℂG ⊗_{ℂH} l^{2}(H)

is a G-equivariant linear map, which is bounded with respect to the norm coming from l^{2}(G). Hence it induces a G-equivariant bounded operator l^{2}(G) → l^{2}(G), which is by definition the image of g ∈ N (H) under N (i).

In the sequel we will ignore the functional analytic aspects of N (G) and will only consider its algebraic properties as a ring.

1.2 Ring Theoretic Properties of the Group von Neumann Algebra

On the first glance the von Neumann algebra N (G) looks not very nice as a ring. It is an integral domain, i.e. has no non-trivial zero-divisors if and only if G is trivial. It is Noetherian if and only if G is finite (see [80, Exercise 9.11]). It is for instance easy to see that N (ℤ^{n}) ≅ L^{∞}(T^{n}) does contain non-trivial zero-divisors and is not Noetherian. The main advantage of N (G) is that it contains many more idempotents than ℂG. This has the effect that N (G) has the following ring theoretic property. A ring R is called semihereditary if every finitely generated submodule of a projective module is again projective. This implies that the category of finitely presented R-modules is an abelian category.

Theorem 1.6 (Von Neumann algebras are semihereditary). Any von Neumann algebra A is semihereditary.

Proof. This follows from the facts that any von Neumann algebra is a Baer ∗-ring and hence in particular a Rickart C^{∗}-algebra [5, Definition 1, Definition 2 and Proposition 9 in Chapter 1.4] and that a C^{∗}-algebra is semihereditary if and only if it is Rickart [1, Corollary 3.7 on page 270].

Remark 1.7 (Group von Neumann algebras are semihereditary). It is quite useful to study the following elementary proof of Theorem 1.6 in the special case of a group von Neumann algebra N (G). One easily checks that it suffices to show for a finitely generated submodule M ⊆ N (G)^{n} that M is projective. Let f: N (G)^{m} → N (G)^{n} be an N (G)-linear map. Choose a matrix A ∈ M(m, n; N (G)) such that f is given by right multiplication with A. Because of N (G) = B(l^{2}(G))^{G} we can define a G-equivariant bounded operator

ν(f): l^{2}(G)^{m} → l^{2}(G)^{n}, (u_{1}, …, u_{m}) ↦ (∑_{i=1}^{m} \overline{a_{i,1}^{∗}(\overline{u_{i}})}, …, ∑_{i=1}^{m} \overline{a_{i,n}^{∗}(\overline{u_{i}})}) ,

where by definition \overline{∑_{g∈G} λ_{g} · g} := ∑_{g∈G} \overline{λ_{g}} · g and a_{i,j}^{∗} denotes the adjoint of a_{i,j}. With these conventions ν(id) = id, ν(r · f + s · g) = r · ν(f) + s · ν(g) and ν(g ∘ f) = ν(g) ∘ ν(f) for r, s ∈ ℂ and N (G)-linear maps f and g. Moreover we have ν(f)^{∗} = ν(f^{∗}) for an N (G)-map f: N (G)^{m} → N (G)^{n}, where f^{∗}: N (G)^{n} → N (G)^{m} is given by right multiplication with the matrix (a_{j,i}^{∗}), if f is given by right multiplication with the matrix (a_{i,j}), and ν(f)^{∗} is the adjoint of the operator ν(f).

Every equivariant bounded operator l^{2}(G)^{m} → l^{2}(G)^{n} can be written as ν(f) for a unique f. Moreover, the sequence N (G)^{m} ⟶^{f} N (G)^{n} ⟶^{g} N (G)^{p} of N (G)- modules is exact if and only if the sequence of bounded G-equivariant operators

l^{2}(G)^{m} ⟶^{ν(f)} l^{2}(G)^{n} ⟶^{ν(g)} l^{2}(G)^{p} is exact. More details and explanations for the last two statements can be found in [80, Section 6.2].

Consider the finitely generated N (G)-submodule M ⊆ N (G)^{n}. Choose an N (G)-linear map f: N (G)^{m} → N (G)^{n} with image M. The kernel of ν(f) is a closed G-invariant linear subspace of l^{2}(G)^{m}. Hence there is an N (G)-map p: N (G)^{m} → N (G)^{m} such that ν(p) is a G-equivariant projection, whose image is ker(ν(f)). Now ν(p) ∘ ν(p) = ν(p) implies p ∘ p = p and im(ν(p)) = ker(ν(f)) implies im(p) = ker(f). Hence ker(f) is a direct summand in N (G)^{m} and im(f) = M is projective.

The point is that in order to get the desired projection p one passes to the interpretation by Hilbert spaces and uses orthogonal projections there. We have enlarged the group ring ℂG to the group von Neumann algebra N (G), which does contain these orthogonal projections in contrast to ℂG.

1.3 Dimension Theory over the Group von Neumann Algebra

An important feature of the group von Neumann algebra is its trace.

Definition 1.8 (Von Neumann trace). The von Neumann trace on N (G) is defined by

tr_{N (G)} : N (G) → ℂ, f ↦ ⟨f(e), e⟩_{l^{2}(G)} ,

where e ∈ G ⊆ l^{2}(G) is the unit element.

It enables us to define a dimension for finitely generated projective N (G)- modules.

Definition 1.9 (Von Neumann dimension for finitely generated projective N (G)-modules). Let P be a finitely generated projective N (G)-module. Choose a matrix A = (a_{i,j}) ∈ M(n, n; N (G)) with A^{2} = A such that the image of the N (G)-linear map r_{A}: N (G)^{n} → N (G)^{n} given by right multiplication with A is N (G)-isomorphic to P. Define the von Neumann dimension of P by

dim_{N (G)}(P) := ∑_{i=1}^{n} tr_{N (G)}(a_{i,i}) ∈ [0, ∞).

We omit the standard proof that dim_{N (G)}(P) depends only on the isomorphism class of P but not on the choice of the matrix A. Obviously

dim_{N (G)}(P ⊕ Q) = dim_{N (G)}(P) + dim_{N (G)}(Q).

It is not hard to show that dim_{N (G)} is faithful, i.e. dim_{N (G)}(P) = 0 ⇔ P = 0 holds for any finitely generated projective N (G)-module P.

Recall that the dual M^{∗} of a left or right R-module M is the right or left R-module hom_{R}(M, R) respectively, where the R-multiplication is given by (fr)(x) = f(x)r or (rf)(x) = rf(x) respectively for f ∈ M^{∗}, x ∈ M and r ∈ R.

Definition 1.10 (Closure of a submodule). Let M be an R-submodule of N. Define the closure of M in N to be the R-submodule of N

\overline{M} := \{x ∈ N | f(x) = 0 for all f ∈ N^{∗} with M ⊆ ker(f)\}.

For an R-module M define the R-submodule TM and the quotient R-module PM by

TM := {x ∈ M | f(x) = 0 for all f ∈ M^{∗}};
PM := M/TM.

Notice that TM is the closure of the trivial submodule in M. It can also be described as the kernel of the canonical map i(M): M → (M^{∗})^{∗}, which sends x ∈ M to the map M^{∗} → R, f ↦ f(x). Notice that TPM = 0, PPM = PM, M^{∗} = (PM)^{∗} and that PM = 0 is equivalent to M^{∗} = 0.

The next result is the key ingredient in the definition of L^{2}-Betti numbers for G-spaces. Its proof can be found in [76, Theorem 0.6], [80, Theorem 6.7].

Theorem 1.11. (Dimension function for arbitrary N (G)-modules).

(i) If K ⊆ M is a submodule of the finitely generated N (G)-module M, then M/\overline{K} is finitely generated projective and \overline{K} is a direct summand in M;

(ii) If M is a finitely generated N (G)-module, then PM is finitely generated projective, there is an exact sequence 0 → N (G)^{n} → N (G)^{n} → TM → 0 and

M ≅ PM ⊕ TM;

(iii) There exists precisely one dimension function

dim_{N (G)} : {N (G)-modules} → [0, ∞] := {r ∈ ℝ | r ≥ 0} ⨿ {∞}

which satisfies:

(a) Extension Property If M is a finitely generated projective N (G)-module, then dim_{N (G)}(M) agrees with the expression introduced in Definition 1.9;

(b) Additivity If 0 → M_{0} → M_{1} → M_{2} → 0 is an exact sequence of N (G)-modules, then

dim_{N (G)}(M_{1}) = dim_{N (G)}(M_{0}) + dim_{N (G)}(M_{2}),

where for r, s ∈ [0, ∞] we define r + s by the ordinary sum of two real numbers if both r and s are not ∞, and by ∞ otherwise; (c) Cofinality

Let {M_{i} | i ∈ I} be a cofinal system of submodules of M, i.e. M = ⋃_{i∈I} M_{i} and for two indices i and j there is an index k in I satisfying M_{i}, M_{j} ⊆ M_{k}. Then

dim_{N (G)}(M) = sup{dim_{N (G)}(M_{i}) | i ∈ I};

(d) Continuity If K ⊆ M is a submodule of the finitely generated N (G)-module M, then

dim_{N (G)}(K) = dim_{N (G)}(\overline{K}).

Definition 1.12 (Von Neumann dimension for arbitrary N (G)-modules). In the sequel we mean for an (arbitrary) N (G)-module M by dim_{N (G)}(M) the value of the dimension function appearing in Theorem 1.11 and call it the von Neumann dimension of M.

Remark 1.13 (Uniqueness of the dimension function). There is only one possible definition for the dimension function appearing in Theorem 1.11, namely one must have

dim_{N (G)}(M) := sup{dim_{N (G)}(P) | P ⊆ M finitely generated projective submodule} ∈ [0, ∞].

Namely, consider the directed system of finitely generated N (G)-submodules {M_{i} | i ∈ I} of M which is directed by inclusion. By Cofinality

dim_{N (G)}(M) = sup{dim_{N (G)}(M_{i}) | i ∈ I}.

From Additivity and Theorem 1.11 (ii) we conclude

dim_{N (G)}(M_{i}) = dim_{N (G)}(PM_{i})

and that PM_{i} is finitely generated projective. This shows uniqueness of dim_{N (G)}. The hard part in the proof of Theorem 1.11 (iii) is to show that the definition above does have all the desired properties.

We also see what dim_{N (G)}(M) = 0 means. It is equivalent to the condition that M contains no non-trivial projective N (G)-submodule, or, equivalently, no non-trivial finitely generated projective N (G)-submodule.

Example 1.14 (The von Neumann dimension for finite groups). If G is finite, then N (G) = ℂG and tr_{N (G)} (∑_{g∈G} λ_{g} · g) is the coefficient λ_{e} of the unit element e ∈ G. For an N (G)-module M its von Neumann dimension dim_{N (G)}(V) is \frac{1}{|G|}-times the complex dimension of the underlying complex vector space M.

The next example implies that dim_{N (G)}(P) for a finitely generated projective N (G)-module can be any non-negative real number.

Example 1.15 (The von Neumann dimension for ℤ^{n}). Consider G = ℤ^{n}. Recall that N (ℤ^{n}) = L^{∞}(T^{n}). Under this identification we get for the von Neumann trace

tr_{N (ℤ^{n})} : N (ℤ^{n}) → ℂ, f ↦ ∫_{T^{n}} fdμ,

where μ is the standard Lebesgue measure on T^{n}.

Let X ⊆ T^{n} be any measurable set and χ_{X} ∈ L^{∞}(T^{n}) be its characteristic function. Denote by M_{χ_{X}}: L^{2}(T^{n}) → L^{2}(T^{n}) the ℤ^{n}-equivariant unitary projection given by multiplication with χ_{X}. Its image P is a finitely generated projective N (ℤ^{n})-module, whose von Neumann dimension dim_{N (ℤ^{n})}(P) is the volume μ(X) of X.

In view of the results above the following slogan makes sense.

Slogan 1.16. The group von Neumann algebra N (G) behaves like the ring of integers ℤ provided one ignores the properties integral domain and Noetherian.

Namely, Theorem 1.11 (ii) corresponds to the statement that a finitely generated ℤ-module M decomposes into M = M/ tors(M) ⊕ tors(M) and that there exists an exact sequence of ℤ-modules 0 → ℤ^{n} → ℤ^{n} → tors(M) → 0, where tors(M) is the ℤ-module consisting of torsion elements. One obtains the obvious analog of Theorem 1.11 (iii) if one considers

{ℤ-modules} → [0, ∞], M ↦ dim_{ℚ}(ℚ ⊗_{ℤ} M).

One basic difference between the case ℤ and N (G) is that there exist projective N (G)-modules with finite dimension which are not finitely generated, which is not true over ℤ. For instance take the direct sum P = ⨁_{i=1}^{∞} P_{i} of N (ℤ^{n})- modules P_{i} appearing in Example 1.15 with dim_{N (ℤ^{n})}(P_{i}) = 2^{−i}. Then P is projective but not finitely generated and satisfies dim_{N (ℤ^{n})}(P) = 1.

The proof of the following two results is given in [80, Theorem 6.13 and Theorem 6.39].

Theorem 1.17 (Dimension and colimits). Let {M_{i} | i ∈ I} be a directed system of N (G)-modules over the directed set I. For i ≤ j let ϕ_{i,j}: M_{i} → M_{j} be the associated morphism of N (G)-modules. For i ∈ I let ψ_{i}: M_{i} → colim_{i∈I} M_{i} be the canonical morphism of N (G)-modules. Then:

(i) We get for the dimension of the N (G)-module given by the colimit

dim_{N (G)} (colim_{i∈I} M_{i}) = sup {dim_{N (G)}(im(ψ_{i})) | i ∈ I};

(ii) Suppose for each i ∈ I that there exists i_{0} ∈ I with i ≤ i_{0} such that dim_{N (G)}(im(ϕ_{i,i_{0}})) < ∞ holds. Then

dim_{N (G)} (colim_{i∈I} M_{i})
= sup {inf {dim_{N (G)}(im(ϕ_{i,j}: M_{i} → M_{j})) | j ∈ I, i ≤ j} | i ∈ I} .

Theorem 1.18 (Induction and dimension). Let i: H → G be an injective group homomorphism. Then

(i) Induction with N (i): N (H) → N (G) is a faithfully flat functor M ↦ i_{∗}M := N (G)⊗_{N (i)} M from the category of N (H)-modules to the category of N (G)-modules, i.e. a sequence of N (H)-modules M_{0} → M_{1} → M_{2} is exact at M_{1} if and only if the induced sequence of N (G)-modules i_{∗}M_{0} → i_{∗}M_{1} → i_{∗}M_{2} is exact at i_{∗}M_{1};

(ii) For any N (H)-module M we have:

dim_{N (H)}(M) = dim_{N (G)}(i_{∗}M).

Example 1.19 (The von Neumann dimension and ℂ[ℤ^{n}]-modules). Consider the case G = ℤ^{n}. Then ℂ[ℤ^{n}] is a commutative integral domain and hence has a quotient field ℂ[ℤ^{n}]_{(0)}. Let dim_{ℂ[ℤ^{n}]_{(0)}} denote the usual dimension for vector spaces over ℂ[ℤ^{n}]_{(0)}. Let M be a ℂ[ℤ^{n}]-module. Then

dim_{N (ℤ^{n})} (N (ℤ^{n}) ⊗_{ℂ[ℤ^{n}]} M) = dim_{ℂ[ℤ^{n}]_{(0)}} (ℂ[ℤ^{n}]_{(0)} ⊗_{ℂ[ℤ^{n}]} M) . (1.20)

This follows from the following considerations. Let {M_{i} | i ∈ I} be the directed system of finitely generated submodules of M. Then M = colim_{i∈I} M_{i}. Since the tensor product has a right adjoint, it is compatible with colimits. This implies together with Theorem 1.17

dim_{N (ℤ^{n})} (N (ℤ^{n}) ⊗_{ℂ[ℤ^{n}]} M) = sup {dim_{N (ℤ^{n})} (N (ℤ^{n}) ⊗_{ℂ[ℤ^{n}]} M_{i})};
dim_{ℂ[ℤ^{n}]_{(0)}} (ℂ[ℤ^{n}]_{(0)} ⊗_{ℂ[ℤ^{n}]} M) = sup {dim_{ℂ[ℤ^{n}]_{(0)}} (ℂ[ℤ^{n}]_{(0)} ⊗_{ℂ[ℤ^{n}]} M_{i})} .

Hence it suffices to prove the claim for a finitely generated ℂ[ℤ^{n}]-module N. The case n = 1 is easy. Then ℂ[ℤ] is a principal integral domain and we can write

N = ℂ[ℤ]^{r} ⊕ ⨁_{i=1}^{k} ℂ[ℤ]/(u_{i})

for non-trivial elements u_{i} ∈ ℂ[ℤ] and some non-negative integers k and r. One easily checks that there is an exact N (ℤ)-sequence

0 → N (ℤ) ⟶^{r_{u_{i}}} N (ℤ) → N (ℤ) ⊗_{ℂ[ℤ]} ℂ[ℤ]/(u_{i}) → 0

using the identification N (ℤ) = L^{∞}(S^{1}) from Example 1.4 to show injectivity of the map r_{u_{i}} given by multiplication with u_{i}. This implies

dim_{N (ℤ)} (N (ℤ) ⊗_{ℂ[ℤ]} N) = r = dim_{ℂ[ℤ]_{(0)}} (ℂ[ℤ]_{(0)} ⊗_{ℂ[ℤ]} N) .

In the general case n ≥ 1 one knows that there exists a finite free ℂ[ℤ^{n}]- resolution of N. Now the claim follows from [80, Lemma 1.34].

This example is the commutative version of a general setup for arbitrary groups, which will be discussed in Subsection 4.2.

A center-valued dimension function for finitely generated projective modules will be introduced in Definition 7.3. It can be used to classify finitely generated projective N (G)-modules (see Theorem 7.5) and shows that the representation theory of finite dimensional representations over a finite group extends to infinite groups if one works with N (G) (see Remark 7.6).

2 Definition and Basic Properties of L^{2}-Betti Numbers

In this section we define L^{2}-Betti numbers for arbitrary G-spaces and study their basic properties. Our general algebraic definition is very general and is very flexible. This allows to apply standard techniques such as spectral sequences and Mayer-Vietoris arguments directly. The original analytic definition for free proper smooth G-manifolds with G-invariant Riemannian metrics is due to Atiyah and will be briefly discussed in Subsection 2.3.

2.1 The Definition of L^{2}-Betti Numbers

Definition 2.1 (L^{2}-Betti numbers of G-spaces). Let X be a (left) G-space. Equip N (G) with the obvious N (G)-ℤG-bimodule structure. The singular homology H_{p}^{G}(X; N (G)) of X with coefficients in N (G) is the homology of the N (G)-chain complex N (G) ⊗_{ℤG} C_{∗}^{sing}(X), where C_{∗}^{sing}(X) is the singular chain complex of X with the induced ℤG-structure. Define the p-th L^{2}-Betti number of X by

b_{p}^{(2)}(X; N (G)) := dim_{N (G)} (H_{p}^{G}(X; N (G))) ∈ [0, ∞],

where dim_{N (G)} is the dimension function of Definition 1.12.

If G and its action on X are clear from the context, we often omit N (G) in the notation above. For instance, for a connected CW -complex X we denote by b_{p}^{(2)}(X̃) the L^{2}-Betti number b_{p}^{(2)}(X̃; N (π_{1}(X))) of its universal covering X̃ with respect to the obvious π_{1}(X)-action.

Notice that we have no assumptions on the G-action or on the topology on X, we do not need to require that the operation is free, proper, simplicial or cocompact. Thus we can apply this definition to the classifying space for free proper G-actions EG, which is a free G-CW -complex which is contractible (after forgetting the group action). Recall that EG is unique up to G-homotopy. Its quotient BG = G\EG is a connected CW -complex, which is up to homotopy uniquely determined by the property that π_{n}(BG) = {1} for n ≥ 2 and π_{1}(BG) ≅ G holds, and called classifying space of G. Moreover, G → EG → BG is the universal G-principal bundle.

Definition 2.2 (L^{2}-Betti numbers of groups). Define for any (discrete) group G its p-th L^{2}-Betti number by

b_{p}^{(2)}(G) := b_{p}^{(2)}(EG, N (G)).

Remark 2.3 (Comparison with the approach by Cheeger and Gromov). A detailed comparison of our approach with the one by Cheeger and Gromov [15, section 2] can be found in [80, Remark 6.76]. Cheeger and Gromov [15, Section 2] define L^{2}-cohomology and L^{2}-Betti numbers of a G-space X by considering the category whose objects are G-maps f: Y → X for a simplicial complex Y with cocompact free simplicial G-action and then using inverse limits to extend the classical notions for finite free G-CW -complexes such as Y to X. Their approach is technically more complicated because for instance they work with cohomology instead of homology and therefore have to deal with inverse limits instead of directed limits. Our approach is closer to standard notions, the only non-standard part is the verification of the properties of the extended dimension function (Theorem 1.11).

Remark 2.4 (L^{2}-Betti numbers for von Neumann algebras). The algebraic approach to L^{2}-Betti numbers of groups as

b_{p}^{(2)}(G) = dim_{N (G)} (Tor_{p}^{ℂG}(ℂ, N (G)))

based on the dimension function for arbitrary modules and homological algebra plays a role in the definition of L^{2}-Betti numbers for certain von Neumann algebras by Connes-Shlyakhtenko [18]. The point of their construction is to introduce invariants which depend on the group von Neumann algebra N (G) only. If one could show that their invariants applied to N (G) agree with the L^{2}-Betti numbers of G, one would get a positive answer to the open problem, whether the von Neumann algebras of two finitely generated free groups F_{1} and F_{2} are isomorphic as von Neumann algebras if and only if the groups F_{1} and F_{2} are isomorphic.

Definition 2.5 (G-CW -complex). A G-CW -complex X is a G-space together with a G-invariant filtration

∅ = X_{−1} ⊆ X_{0} ⊆ X_{1} ⊆ … ⊆ X_{n} ⊆ … ⊆ ⋃_{n≥0} X_{n} = X

such that X carries the colimit topology with respect to this filtration (i.e. a set C ⊆ X is closed if and only if C ∩ X_{n} is closed in X_{n} for all n ≥ 0) and X_{n} is obtained from X_{n−1} for each n ≥ 0 by attaching equivariant n-dimensional cells, i.e. there exists a G-pushout

∐_{i∈I_{n}} G/H_{i} × S^{n−1} \underrightarrow{∐_{i∈In} q_{i}} X_{n−1}
↓  ↓
∐_{i∈I_{n}} G/H_{i} × D^{n} \overrightarrow{∐_{i∈In} Q_{i}} X_{n}

The space X_{n} is called the n-skeleton of X. A G-CW -complex X is proper if and only if all its isotropy groups are finite. A G-space is called cocompact if G\X is compact. A G-CW -complex X is finite if X has only finitely many equivariant cells. A G-CW -complex is finite if and only if it is cocompact. A G-CW -complex X is of finite type if each n-skeleton is finite. It is called of dimension ≤ n if X = X_{n} and finite dimensional if it is of dimension ≤ n for some integer n. A free G-CW -complex X is the same as a regular covering X → Y of a CW -complex Y with G as group of deck transformations.

Notice that Definition 2.5 also makes sense in the case where G is a topological group. Every proper smooth cocompact G-manifold is a proper G-CW - complex by means of an equivariant triangulation.

For a G-CW -complex one can use the cellular ℤG-chain complex instead of the singular chain complex in the definition of L^{2}-Betti numbers by the next result. Its proof can be found in [76, Lemma 4.2]. For more information about G-CW -complexes we refer for instance to [104, Sections II.1 and II.2], [71, Sections 1 and 2], [80, Subsection 1.2.1].

Lemma 2.6. Let X be a G-CW -complex. Let C_{∗}^{c}(X) be its cellular ℤG-chain complex. Then there is a ℤG-chain homotopy equivalence C_{∗}^{sing}(X) → C_{∗}^{c}(X) and we get

b_{p}^{(2)}(X; N (G)) = dim_{N (G)} (H_{p} (N (G) ⊗_{ℤG} C_{∗}^{c}(X))) .

The definition of b_{p}^{(2)}(X; N (G)) and the above lemma extend in the obvious way to pairs (X, A).

2.2 Basic Properties of L^{2}-Betti Numbers

The basic properties of L^{2}-Betti numbers are summarized in the following theorem. Its proof can be found in [80, Theorem 1.35 and Theorem 6.54] except for assertion (viii) which follows from [80, Lemma 13.45].

Theorem 2.7 (L^{2}-Betti numbers for arbitrary spaces).

(i) Homology invariance We have for a G-map f: X → Y:

(a) Suppose for n ≥ 1 that for each subgroup H ⊆ G the induced map f^{H}: X^{H} → Y^{H} is ℂ-homologically n-connected, i.e. the map

H_{p}^{sing}(f^{H}; ℂ): H_{p}^{sing}(X^{H}; ℂ) → H_{p}^{sing}(Y^{H}; ℂ)

induced by f^{H} on singular homology with complex coefficients is bijective for p < n and surjective for p = n. Then

b_{p}^{(2)}(X) = b_{p}^{(2)}(Y)  for p < n;
b_{p}^{(2)}(X) ≥ b_{p}^{(2)}(Y)  for p = n;

(b) Suppose that for each subgroup H ⊆ G the induced map f^{H}: X^{H} → Y^{H} is a ℂ-homology equivalence, i.e. H_{p}^{sing}(f^{H}; ℂ) is bijective for p ≥ 0. Then

b_{p}^{(2)}(X) = b_{p}^{(2)}(Y) for p ≥ 0;

(ii) Comparison with the Borel construction Let X be a G-CW -complex. Suppose that for all x ∈ X the isotropy group G_{x} is finite or satisfies b_{p}^{(2)}(G_{x}) = 0 for all p ≥ 0. Then

b_{p}^{(2)}(X; N (G)) = b_{p}^{(2)}(EG × X; N (G))  for p ≥ 0,

where G acts diagonally on EG × X;

(iii) Invariance under non-equivariant ℂ-homology equivalences Suppose that f: X → Y is a G-equivariant map of G-CW -complexes such that the induced map H_{p}^{sing}(f; ℂ) on singular homology with complex coefficients is bijective for all p. Suppose that for all x ∈ X the isotropy group G_{x} is finite or satisfies b_{p}^{(2)}(G_{x}) = 0 for all p ≥ 0, and analogously for all y ∈ Y. Then we have for all p ≥ 0

b_{p}^{(2)}(X; N (G)) = b_{p}^{(2)}(Y; N (G));

(iv) Independence of equivariant cells with infinite isotropy Let X be a G-CW -complex. Let X[∞] be the G-CW -subcomplex consisting of those points whose isotropy subgroups are infinite. Then we get for all p ≥ 0

b_{p}^{(2)}(X; N (G)) = b_{p}^{(2)}(X, X[∞]; N (G));

(v) Künneth formula Let X be a G-space and Y be an H-space. Then X × Y is a G × H-space and we get for all n ≥ 0

b_{n}^{(2)}(X × Y) = ∑_{p+q=n} b_{p}^{(2)}(X) · b_{q}^{(2)}(Y),

where we use the convention that 0 · ∞ = 0, r · ∞ = ∞ for r ∈ (0, ∞] and r + ∞ = ∞ for r ∈ [0, ∞];

(vi) Induction Let i: H → G be an inclusion of groups and let X be an H-space. Let N (i): N (H) → N (G) be the induced ring homomorphism (see (1.5)). Then:

H_{p}^{G}(G ×_{H} X; N (G)) = N (G) ⊗_{N (i)} H_{p}^{H}(X; N (H));
b_{p}^{(2)}(G ×_{H} X; N (G)) = b_{p}^{(2)}(X; N (H));

(vii) Restriction to subgroups of finite index Let H ⊆ G be a subgroup of finite index [G : H]. Let X be a G-space and let res_{G}^{H} X be the H-space obtained from X by restriction. Then

b_{p}^{(2)}(res_{G}^{H} X; N (H)) = [G : H] · b_{p}^{(2)}(X; N (G));

(viii) Restriction with epimorphisms with finite kernel Let p: G → Q be an epimorphism of groups with finite kernel K. Let X be a Q-space. Let p^{∗}X be the G-space obtained from X using p. Then

b_{p}^{(2)}(p^{∗}X; N (G)) = \frac{1}{|K|} · b_{p}^{(2)}(X; N (Q));

(ix) Zero-th homology and L^{2}-Betti number Let X be a path-connected G-space. Then:

(a) There is an N (G)-isomorphism H_{0}^{G}(X; N (G)) ⟶^{≅} N (G) ⊗_{ℂG} ℂ;

(b) b_{0}^{(2)}(X; N (G)) = |G|^{−1}, where |G|^{−1} is defined to be zero if the order |G| of G is infinite;

(x) Euler-Poincaré formula Let X be a free finite G-CW -complex. Let χ(G\X) be the Euler characteristic of the finite CW -complex G\X, i.e.

χ(G\X) := ∑_{p≥0}(−1)^{p} · |I_{p}(G\X)|  ∈ ℤ,

where |I_{p}(G\X)| is the number of p-cells of G\X. Then

χ(G\X) = ∑_{p≥0}(−1)^{p} · b_{p}^{(2)}(X);

(xi) Morse inequalities Let X be a free G-CW -complex of finite type. Then we get for n ≥ 0

∑_{p=0}^{n}(−1)^{n−p} · b_{p}^{(2)}(X) ≤ ∑_{p=0}^{n}(−1)^{n−p} · |I_{p}(G\X)|;

(xii) Poincaré duality Let M be a cocompact free proper G-manifold of dimension n which is orientable. Then

b_{p}^{(2)}(M) = b_{n−p}^{(2)}(M, ∂M);

(xiii) Wedges Let X_{1}, X_{2}, …, X_{r} be connected (pointed) CW -complexes of finite type and X = ⋁_{i=1}^{r} X_{i} be their wedge. Then

b_{1}^{(2)}(X̃) − b_{0}^{(2)}(X̃) = r − 1 + ∑_{j=1}^{r} (b_{1}^{(2)}(X̃_{j}) − b_{0}^{(2)}(X̃_{j}));

b_{p}^{(2)}(X̃) = ∑_{j=1}^{r} b_{p}^{(2)}(X̃_{j})  for 2 ≤ p;

(xiv) Connected sums Let M_{1}, M_{2}, …, M_{r} be compact connected m-dimensional manifolds for m ≥ 3. Let M be their connected sum M_{1}# … #M_{r}. Then

b_{1}^{(2)}(M̃) − b_{0}^{(2)}(M̃) = r − 1 + ∑_{j=1}^{r} (b_{1}^{(2)}(M̃_{j}) − b_{0}^{(2)}(M̃_{j}));

b_{p}^{(2)}(M̃) = ∑_{j=1}^{r} b_{p}^{(2)}(M̃_{j})  for 2 ≤ p ≤ m − 2.

Example 2.8. If G is finite, then b_{p}^{(2)}(X; N (G)) reduces to the classical Betti number b_{p}(X) multiplied with the factor |G|^{−1}.

Remark 2.9 (Reading off L^{2}-Betti numbers from H_{p}(X; ℂ)). If f: X → Y is a G-map of free G-CW -complexes which induces isomorphisms H_{p}^{sing}(f; ℂ) for all p ≥ 0, then Theorem 2.7 (i) implies

b_{p}^{(2)}(X; N (G)) = b_{p}^{(2)}(Y; N (G)).

This does not necessarily mean that one can read off b_{p}^{(2)}(X; N (G)) from the singular homology H_{p}(X; ℂ) regarded as a ℂG-module in general. In general there is for a free G-CW -complex X a spectral sequence converging to H_{p+q}^{G}(X; N (G)), whose E^{2}-term is

E_{p,q}^{2} = Tor_{p}^{ℂG}(H_{q}(X; ℂ), N (G)).

There is no reason why the equality of the dimension of the E^{2}-term for two free G-CW -complexes X and Y implies that the dimension of H_{p+q}^{G}(X; N (G)) and H_{p+q}^{G}(Y; N (G)) agree. However, this is the case if the spectral sequence collapses from the dimension point of view. For instance, if we make the assumption dim_{N (G)} (Tor_{p}^{ℂG}(M, N (G))) = 0 for all ℂG-modules M and p ≥ 2, Additivity and Cofinality of dim_{N (G)} (see Theorem 1.11) imply

b_{p}^{(2)}(X; N (G)) =

dim_{N (G)} (N (G) ⊗_{ℂG} H_{p}(X; ℂ)) + dim_{N (G)} (Tor_{1}^{ℂG}(H_{p−1}(X; ℂ), N (G))) .

The assumption above is satisfied if G is amenable (see Theorem 5.1) or G has cohomological dimension ≤ 1 over ℂ, for instance, if G is virtually free.

Remark 2.10 (L^{2}-Betti numbers ignore infinite isotropy). Theorem 2.7 (iv) says that the L^{2}-Betti numbers do not see the part of a G-space X whose isotropy groups are infinite. In particular b_{p}^{(2)}(X; N (G)) = 0 if X is a G-CW - complex whose isotropy groups are all infinite. This follows from the fact that for a subgroup H ⊆ G

dim_{N (G)} (N (G) ⊗_{ℂG} ℂ[G/H]) = \{ \begin{matrix}\frac{1}{|H|} & if |H| < ∞; \\ 0 & if |H| = ∞. \end{matrix}

Remark 2.11 (L^{2}-Betti numbers often vanish). An important phenomenon is that the L^{2}-Betti numbers of universal coverings of spaces and of groups tend to vanish more often than the classical Betti numbers. This allows to draw interesting conclusions as we will see later.

2.3 Comparison with Other Definitions

In this subsection we give a short overview of the previous definitions of L^{2}-Betti numbers. Originally they were defined in terms of heat kernels. Their analytic aspects are important, but we will only focus on their algebraic aspects in this survey article. So a reader may skip the brief explanations below.

The notion of L^{2}-Betti numbers is due to Atiyah [2]. He defined for a smooth Riemannian manifold with a free proper cocompact G-action by isometries its analytic p-th L^{2}-Betti number by the following expression in terms of the heat kernel e^{−tΔ_{p}}(x, y) of the p-th Laplacian Δ_{p}

b_{p}^{(2)}(M) = lim_{t→∞} ∫_{F} tr_{ℂ}(e^{−tΔ_{p}}(x, x)) dvol_{x},  (2.12)

where F is a fundamental domain for the G-action and tr_{ℂ} denotes the trace of an endomorphism of a finite-dimensional vector space. The L^{2}-Betti numbers are invariants of the large times asymptotic of the heat kernel.

A finitely generated Hilbert N (G)-module is a Hilbert space V together with a linear G-action by isometries such that there exists a linear isometric Gembedding into l^{2}(G)^{n} for some n ≥ 0. One can assign to it its von Neumann dimension by

dim_{N (G)}(V) := tr_{N (G)}(A) ∈ [0, ∞),

where A is any idempotent matrix A ∈ M(n, n; N (G)) such that the image of the G-equivariant operator l^{2}(G)^{n} → l^{2}(G)^{n} induced by A is isometrically linearly G-isomorphic to V.

The expression in (2.12) can be interpreted as the von Neumann dimension of the space H_{(2)}^{p}(M) of square-integrable harmonic p-forms on M, which is a finitely generated Hilbert N (G)-module (see [2, Proposition 4.16 on page 63])

lim_{t→∞} ∫_{F} tr_{ℂ}(e^{−tΔ_{p}}(x, x)) dvol_{x} = dim_{N (G)} (H_{(2)}^{p}(M)) .  (2.13)

Given a cocompact free G-CW -complex X, one obtains a chain complex of finitely generated Hilbert N (G)-modules C_{∗}^{(2)}(X) := C_{∗}^{c}(X) ⊗_{ℤG} l^{2}(G). Its reduced p-th L^{2}-homology is the finitely generated Hilbert N (G)-module

H_{p}^{(2)}(X; l^{2}(G)) = ker(c_{p}^{(2)})/\overline{im(c_{p+1}^{(2)})}.  (2.14)

Notice that we divide out the closure of the image of the (p + 1)-th differential c_{p+1}^{(2)} of C_{∗}^{(2)}(X) in order to ensure that we obtain a Hilbert space. Then by a result of Dodziuk [24] there is an isometric bijective G-operator

H_{(2)}^{p}(M) ⟶^{≅} H_{p}^{(2)}(K; l^{2}(G)),  (2.15)

where K is an equivariant triangulation of M. Finally one can show [74, Theorem 6.1]

b_{p}^{(2)}(K; N (G)) = dim_{N (G)} (H_{p}^{(2)}(K; l^{2}(G))) ,  (2.16)

where b_{p}^{(2)}(K; N (G)) is the p-th L^{2}-Betti number in the sense of Definition 2.1.

All in all we see that our Definition 2.1 of L^{2}-Betti numbers for arbitrary G-spaces extends the heat kernel definition of (2.12) for smooth Riemannian manifolds with a free proper cocompact G-action by isometries. More details of all these definitions and of their identifications can be found in [80, Chapter 1].

2.4 L^{2}-Euler Characteristic

In this section we introduce the notion of L^{2}-Euler characteristic.

If X is a G-CW -complex, denote by I(X) the set of its equivariant cells. For a cell c ∈ I(X) let (G_{c}) be the conjugacy class of subgroups of G given by its orbit type and let dim(c) be its dimension. Denote by |G_{c}|^{−1} the inverse of the order of any representative of (G_{c}), where |G_{c}|^{−1} is to be understood to be zero if the order is infinite.

Definition 2.17 (L^{2}-Euler characteristic). Let G be a group and let X be a G-space. Define

h^{(2)}(X; N (G)) := ∑_{p≥0} b_{p}^{(2)}(X; N (G)) ∈ [0, ∞];
χ^{(2)}(X; N (G)) := ∑_{p≥0}(−1)^{p} · b_{p}^{(2)}(X; N (G)) ∈ ℝ, if h^{(2)}(X; N (G)) < ∞;
m(X; G)  := ∑_{c∈I(X)} |G_{c}|^{−1} ∈ [0, ∞], if X is a G-CW -complex;
h^{(2)}(G)  := h^{(2)}(EG; N (G)) ∈ [0, ∞];
χ^{(2)}(G)  := χ^{(2)}(EG; N (G)) ∈ ℝ, if h^{(2)}(G) < ∞.

We call χ^{(2)}(X; N (G)) and χ^{(2)}(G) the L^{2}-Euler characteristic of X and G.

The condition h^{(2)}(X; N (G)) < ∞ ensures that the sum which appears in the definition of χ^{(2)}(X; N (G)) converges absolutely and that the following results are true. The reader should compare the next theorem with [15, Theorem 0.3 on page 191]. It essentially follows from Theorem 2.7. Details of its proof can be found in [80, Theorem 6.80].

Theorem 2.18 (L^{2}-Euler characteristic).

(i) Generalized Euler-Poincaré formula Let X be a G-CW -complex with m(X; G) < ∞. Then

h^{(2)}(X; N (G)) < ∞;
∑_{c∈I(X)}(−1)^{dim(c)} · |G_{c}|^{−1} = χ^{(2)}(X; N (G));

(ii) Sum formula Consider the following G-pushout

X_{0} ⟶^{i_{1}} X_{1}
i_{2}↓  ↓^{j_{1}}
X_{2} ⟶_{j_{2}} X

such that i_{1} is a G-cofibration. Suppose that h^{(2)}(X_{i}; N (G)) < ∞ for i = 0, 1, 2. Then

h^{(2)}(X; N (G)) < ∞;
χ^{(2)}(X; N (G)) = χ^{(2)}(X_{1}; N (G)) + χ^{(2)}(X_{2}; N (G)) − χ^{(2)}(X_{0}; N (G));

(iii) Comparison with the Borel construction Let X be a G-CW -complex. If for all c ∈ I(X) the group G_{c} is finite or b_{p}^{(2)}(G_{c}) = 0 for all p ≥ 0, then

b_{p}^{(2)}(X; N (G)) = b_{p}^{(2)}(EG × X; N (G))  for p ≥ 0;
h^{(2)}(X; N (G)) = h^{(2)}(EG × X; N (G));
χ^{(2)}(X; N (G)) = χ^{(2)}(EG × X; N (G)), if h^{(2)}(X; N (G)) < ∞;
∑_{c∈I(X)}(−1)^{dim(c)} · |G_{c}|^{−1} = χ^{(2)}(EG × X; N (G)), if m(X; G) < ∞;

(iv) Invariance under non-equivariant ℂ-homology equivalences Suppose that f: X → Y is a G-equivariant map of G-CW -complexes with m(X; G) < ∞ and m(Y; G) < ∞, such that the induced map H_{p}(f; ℂ) on homology with complex coefficients is bijective for all p ≥ 0. Suppose that for all c ∈ I(X) the group G_{c} is finite or b_{p}^{(2)}(G_{c}) = 0 for all p ≥ 0, and analogously for all d ∈ I(Y). Then

χ^{(2)}(X; N (G)) = ∑_{c∈I(X)}(−1)^{dim(c)} · |G_{c}|^{−1}
= ∑_{d∈I(Y)}(−1)^{dim(d)} · |G_{d}|^{−1}
= χ^{(2)}(Y; N (G));

(v) Künneth formula Let X be a G-CW -complex and Y be an H-CW -complex. Then we get for the G × H-CW -complex X × Y

m(X × Y; G × H) = m(X; G) · m(Y; H);
h^{(2)}(X × Y; N (G × H)) = h^{(2)}(X; N (G)) · h^{(2)}(Y; N (H));
χ^{(2)}(X × Y; N (G × H)) = χ^{(2)}(X; N (G)) · χ^{(2)}(Y; N (H)),

if h^{(2)}(X; N (G)), h^{(2)}(Y; N (H)) < ∞,

where we use the convention that 0 · ∞ = 0 and r · ∞ = ∞ for r ∈ (0, ∞];

(vi) Induction Let H ⊆ G be a subgroup and let X be an H-space. Then

m(G ×_{H} X; G) = m(X; H);
h^{(2)}(G ×_{H} X; N (G)) = h^{(2)}(X; N (H));
χ^{(2)}(G ×_{H} X; N (G)) = χ^{(2)}(X; N (H)),  if h^{(2)}(X; N (H)) < ∞;

(vii) Restriction to subgroups of finite index Let H ⊆ G be a subgroup of finite index [G : H]. Let X be a G-space and let res_{G}^{H} X be the H-space obtained from X by restriction. Then

m(res_{G}^{H} X; H) = [G : H] · m(X; G);
h^{(2)}(res_{G}^{H} X; N (H)) = [G : H] · h^{(2)}(X; N (G));
χ^{(2)}(res_{G}^{H} X; N (H)) = [G : H] · χ^{(2)}(X; N (G)), if h^{(2)}(X; N (G)) < ∞,

where [G : H] · ∞ is understood to be ∞;

(viii) Restriction with epimorphisms with finite kernel Let p: G → Q be an epimorphism of groups with finite kernel K. Let X be a Q-space. Let p^{∗}X be the G-space obtained from X using p. Then

m(p^{∗}X; G) = |K|^{−1} · m(X; Q);
h^{(2)}(p^{∗}X; N (G)) = |K|^{−1} · h^{(2)}(X; N (Q));
χ^{(2)}(p^{∗}X; N (G)) = |K|^{−1} · χ^{(2)}(X; N (Q)), if h^{(2)}(X; N (Q)) < ∞.

Remark 2.19 (L^{2}-Euler characteristic and virtual Euler characteristic). The L^{2}-Euler characteristic generalizes the notion of the virtual Euler characteristic. Let X be a CW -complex which is virtually homotopy finite, i.e. there is a d-sheeted covering p: \overline{X} → X for some positive integer d such that \overline{X} is homotopy equivalent to a finite CW -complex. Define the virtual Euler characteristic following Wall [105]

χ_{virt}(X) := \frac{χ(\overline{X})}{d}.

One easily checks that this is independent of the choice of p: \overline{X} → X since the classical Euler characteristic is multiplicative under finite coverings. Moreover, we conclude from Theorem 2.18 (i) and (vii) that for virtually homotopy finite X

m(X̃; π_{1}(X)) < ∞;
χ^{(2)}(X̃; N (π_{1}(X))) = χ_{virt}(X).

Remark 2.20 (L^{2}-Euler characteristic and orbifold Euler characteristic). If X is a finite G-CW -complex, then ∑_{c∈I(X)}(−1)^{dim(c)} · |G_{c}|^{−1} is also called orbifold Euler characteristic and agrees with the L^{2}-Euler characteristic by Theorem 2.18 (i).

3 Computations of L^{2}-Betti Numbers

In this section we state some cases where the L^{2}-Betti numbers b_{p}^{(2)}(X̃) for certain compact manifolds or finite CW -complexes X can explicitly be computed. These computations give evidence for certain conjectures such as the Atiyah Conjecture 4.1 for (G, d, ℚ) and the Singer Conjecture 9.1 which we will discuss later. Sometimes we will also make a few comments on their proofs in order to give some insight into the methods. Besides analytic methods, which will not be discussed, standard techniques from topology and algebra such as spectral sequences and Mayer-Vietoris sequences will play a role. With our algebraic setup and the nice properties of the dimension function such as Additivity and Cofinality these tools are directly available, whereas in the original settings, which we have briefly discussed in Subsection 2.3, these methods do not apply directly and, if at all, only after some considerable technical efforts.

3.1 Abelian Groups

Let X be a ℤ^{n}-space. Then we get from (1.20)

b_{p}^{(2)}(X; N (ℤ^{n})) = dim_{ℂ[ℤ^{n}]_{(0)}} (ℂ[ℤ^{n}]_{(0)} ⊗_{ℂ[ℤ^{n}]} H_{p}^{sing}(X; ℂ)) . (3.1)

Notice that b_{p}^{(2)}(X; N (ℤ^{n})) is always an integer or ∞.

3.2 Finite Coverings

Let p: X → Y be a finite covering with d-sheets. Then we conclude from Theorem 2.7 (vii)

b_{p}^{(2)}(X̃) = d · b_{p}^{(2)}(Ỹ).  (3.2)

This implies for every connected CW -complex X which admits a selfcovering X → X with d-sheets for d ≥ 2 that b_{p}^{(2)}(X̃) = 0 for all p ∈ ℤ. In particular

b_{p}^{(2)}(S̃^{1}) = 0 for all p ∈ ℤ.  (3.3)

3.3 Surfaces

Let F_{g}^{d} be the orientable closed surface of genus g with d embedded 2-disks removed. (As any non-orientable compact surface is finitely covered by an orientable surface, it suffices to handle the orientable case by (3.2).) From the value of the zero-th L^{2}-Betti number, the Euler-Poincaré formula and Poincaré duality (see Theorem 2.7 (ix), (x) and (xii)) and from the fact that a compact surface with boundary is homotopy equivalent to a bouquet of circles, we conclude

b_{0}^{(2)}(F̃_{g}^{d}) = \{ \begin{matrix}1 & if g = 0, d = 0, 1; \\ 0 & otherwise; \end{matrix}
b_{1}^{(2)}(F̃_{g}^{d}) = \{ \begin{matrix}0 & if g = 0, d = 0, 1; \\ d + 2 · (g − 1) & otherwise; \end{matrix}
b_{2}^{(2)}(F̃_{g}^{d}) = \{ \begin{matrix}1 & if g = 0, d = 0; \\ 0 & otherwise. \end{matrix}

Of course b_{p}^{(2)}(F̃_{g}^{d}) = 0 for p ≥ 3.

3.4 Three-Dimensional Manifolds

In this subsection we state the values of the L^{2}-Betti numbers of compact orientable 3-manifolds.

We begin with collecting some basic notations and facts about 3-manifolds. In the sequel 3-manifold means connected compact orientable 3-manifold, possibly with boundary. A 3-manifold M is prime if for any decomposition of M as a connected sum M_{1}#M_{2}, M_{1} or M_{2} is homeomorphic to S^{3}. It is irreducible if every embedded 2-sphere bounds an embedded 3-disk. Every prime 3-manifold is either irreducible or is homeomorphic to S^{1} × S^{2} [50, Lemma 3.13]. A 3manifold M has a prime decomposition, i.e. one can write M as a connected sum

M = M_{1}#M_{2}# … #M_{r},

where each M_{j} is prime, and this prime decomposition is unique up to renumbering and orientation preserving homeomorphism [50, Theorems 3.15, 3.21].

Recall that a connected CW -complex is called aspherical if π_{n}(X) = 0 for n ≥ 2, or, equivalently, if X̃ is contractible. Any aspherical 3-manifold is homotopy equivalent to an irreducible 3-manifold with infinite fundamental group or to a 3-disk. By the Sphere Theorem [50, Theorem 4.3], an irreducible 3-manifold is aspherical if and only if it is a 3-disk or has infinite fundamental group.

Let us say that a prime 3-manifold is exceptional if it is closed and no finite covering of it is homotopy equivalent to a Haken, Seifert or hyperbolic 3-manifold. No exceptional prime 3-manifolds are known. Both Thurston’s Geometrization Conjecture and Waldhausen’s Conjecture that any 3-manifold is finitely covered by a Haken manifold imply that there are none.

Details of the proof of the following theorem can be found in [69, Sections 5 and 6]. The proof is quite interesting since it uses both topological and analytic tools and relies on Thurston’s Geometrization.

Theorem 3.4 (L^{2}-Betti numbers of 3-manifolds). Let M be the connected sum M_{1}# … #M_{r} of (compact connected orientable) prime 3-manifolds M_{j} which are non-exceptional. Assume that π_{1}(M) is infinite. Then the L^{2}-Betti numbers of the universal covering M̃ are given by

b_{0}^{(2)}(M̃) = 0;
b_{1}^{(2)}(M̃) = (r − 1) − ∑_{j=1}^{r} \frac{1}{| π_{1}(M_{j}) |} + |\{C ∈ π_{0}(∂M) | C ≅ S^{2}\}| − χ(M);

b_{2}^{(2)}(M̃) = (r − 1) − ∑_{j=1}^{r} \frac{1}{| π_{1}(M_{j}) |} + |\{C ∈ π_{0}(∂M) | C ≅ S^{2}\}|;
b_{3}^{(2)}(M̃) = 0.

Notice that in the situation of Theorem 3.4 the p-th L^{2}-Betti number b_{p}(M̃) is a rational number. It is an integer, if π_{1}(M) is torsion-free, and vanishes, if M is aspherical.

3.5 Symmetric Spaces

Let L be a connected semisimple Lie group with finite center such that its Lie algebra has no compact ideal. Let K ⊆ L be a maximal compact subgroup. Then the manifold M := L/K equipped with a left L-invariant Riemannian metric is a symmetric space of non-compact type with L = Isom(M)^{0} and K = Isom(M)_{x}^{0}, where Isom(M)^{0} is the identity component of the group of isometries Isom(M) and Isom(M)_{x}^{0} is the isotropy group of some point x ∈ M under the Isom(M)^{0}-action. Every symmetric space M of non-compact type can be written in this way. The space M is diffeomorphic to ℝ^{n}. Define its fundamental rank

f-rk(M) := rk_{ℂ}(L) − rk_{ℂ}(K),

where rk_{ℂ}(L) and rk_{ℂ}(K) denotes the so called complex rank of the Lie algebra of L and K respectively (see [62, page 128f]). For a compact Lie group K this is the same as the dimension of a maximal torus. The proof of the next result is due to Borel [6].

Theorem 3.5 (L^{2}-Betti numbers of symmetric spaces of non-compact type). Let M be a closed Riemannian manifold whose universal covering M̃ is a symmetric space of non-compact type.

Then b_{p}^{(2)}(M̃) ≠ 0 if and only if f-rk(M̃) = 0 and 2p = dim(M). If f-rk(M̃) = 0, then dim(M) is even and for 2p = dim(M) we get

0 < b_{p}^{(2)}(M̃) = (−1)^{p} · χ(M).

This applies in particular to a hyperbolic manifold and thus we get the result of Dodziuk [25].

Theorem 3.6. Let M be a hyperbolic closed Riemannian manifold of dimension n. Then

b_{p}^{(2)}(M̃) \{ \begin{matrix}= 0 & if 2p ≠ n \\ > 0 & if 2p = n \end{matrix} .

If n is even, then

(−1)^{n/2} · χ(M) > 0.

The strategy of the proof of Theorem 3.6 is the following. Because of the Euler-Poincaré formula (see Theorem 2.7 (x)) it suffices to show that b_{p}^{(2)}(M̃) = 0 for 2p ≠ n and b_{p}^{(2)}(M̃) > 0 for 2p = n. Because of the Hodge-deRham Theorem (see (2.15)) and the facts that the von Neumann dimension is faithful and M̃ is isometrically diffeomorphic to the hyperbolic space ℍ^{n}, it remains to show that the space of harmonic L^{2}-integrable forms H_{(2)}^{p}(ℍ^{n}) is trivial for 2p ≠ n and non-trivial for 2p = n. Notice that this question is independent of M or the π_{1}(M)-action. Using the rotational symmetry of ℍ^{n}, this question is answered positively by Dodziuk [25].

More generally one has the following so called Proportionality Principle (see [80, Theorem 3.183].)

Theorem 3.7 (Proportionality Principle for L^{2}-Betti numbers). Let M be a simply connected Riemannian manifold. Then there are constants B_{p}^{(2)}(M) for p ≥ 0 depending only on the Riemannian manifold M with the following property: For every discrete group G with a cocompact free proper action on M by isometries the following holds

b_{p}^{(2)}(M; N (G)) = B_{p}^{(2)}(M) · vol(G\M).

3.6 Spaces with S^{1}-Action

The next two theorems are taken from [80, Corollary 1.43 and Theorem 6.65].

Theorem 3.8. (L^{2}-Betti numbers and S^{1}-actions). Let X be a connected S^{1}-CW -complex. Suppose that for one orbit S^{1}/H (and hence for all orbits) the inclusion into X induces a map on π_{1} with infinite image. (In particular the S^{1}-action has no fixed points.)

Then we get

b_{p}^{(2)}(X̃) = 0 for p ∈ ℤ;
χ(X) = 0.

Proof. We give an outline of the idea of the proof in the case where X is a cocompact S^{1}-CW -complex, because it is a very illuminating example. The proof in the general case is given in [80, Theorem 6.65]. It is useful to show the following slightly more general statement that for any finite S^{1}-CW -complex Y and S^{1}-map f: Y → X we get b_{p}^{(2)}(f^{∗}X̃; N (π_{1}(X))) = 0 for all p ≥ 0, where f^{∗}X̃ → Y is the pullback of the universal covering X̃ → X with f. We prove the latter statement by induction over the dimension and the number of S^{1}- equivariant cells in top dimension of Y. In the induction step we can assume that Y is an S^{1}-pushout

S^{1}/H × S^{n−1} ⟶^{q} Z
↓  ↓^{j}
S^{1}/H × D^{n} ⟶_{Q} Y

for n = dim(Y). It induces a pushout of free finite π_{1}(X)-CW -complexes

q^{∗}j^{∗}f^{∗}X̃ ⟶ j^{∗}f^{∗}X̃
↓  ↓
Q^{∗}f^{∗}X̃ ⟶ f^{∗}X̃

The associated long exact Mayer-Vietoris sequence looks like

… H_{p}(q^{∗}f^{∗}X̃; N (π_{1}(X)))

→ H_{p}(Q^{∗}f^{∗}X̃; N (π_{1}(X))) ⊕ H_{p}(j^{∗}f^{∗}X̃; N (π_{1}(X)))
→ H_{p}(f^{∗}X̃; N (π_{1}(X))) → H_{p−1}(q^{∗}j^{∗}f^{∗}X̃; N (π_{1}(X)))

→ H_{p−1}(Q^{∗}f^{∗}X̃; N (π_{1}(X))) ⊕ H_{p−1}(j^{∗}f^{∗}X̃; N (π_{1}(X))) → …

Because of the Additivity of the dimension (see Theorem 1.11 (iii)b) it suffices to prove for all p ∈ ℤ

dim_{N (π_{1}(X))} (H_{p}(j^{∗}f^{∗}X̃; N (π_{1}(X)))) = 0;
dim_{N (π_{1}(X))} (H_{p}(q^{∗}f^{∗}X̃; N (π_{1}(X)))) = 0;
dim_{N (π_{1}(X))} (H_{p}(Q^{∗}f^{∗}X̃; N (π_{1}(X)))) = 0.

The induction hypothesis applies to f ∘j: Z → X and f ∘j ∘q: S^{1}/H ×S^{n−1} → X. Hence it remains to show

dim_{N (π_{1}(X))} (H_{p}(Q^{∗}f^{∗}X̃; N (π_{1}(X)))) = 0.

By elementary covering theory Q^{∗}f^{∗}X̃ is π_{1}(X)-homeomorphic to π_{1}(X) ×_{j}

{ \atop S^{1}/H} × D^{n} for the injective group homomorphism j: π_{1}(S^{1}/H) → π_{1}(X) induced by f ∘ Q. We conclude from the Künneth formula and the compatibility of dimension and induction (see Theorem 2.7 (v) and (vi))

dim_{N (π_{1}(X))} (H_{p}(Q^{∗}f^{∗}X̃; N (π_{1}(X)))) = b_{p}^{(2)}({ \atop S^{1}/H}).

Since S^{1}/H is homeomorphic to S^{1}, we get b_{p}^{(2)}({ \atop S^{1}/H}) = 0 from (3.3). The next result is taken from [80, Corollary 1.43].

Theorem 3.9. Let M be an aspherical closed manifold with non-trivial S^{1}- action. (Non-trivial means that sx ≠ x holds for at least one element s ∈ S^{1} and one element x ∈ M). Then the action has no fixed points and the inclusion of any orbit into X induces an injection on the fundamental groups. All L^{2}-Betti numbers b_{p}^{(2)}(M̃) are trivial and χ(M) = 0.

3.7 Mapping Tori

Let f: X → X be a selfmap. Its mapping torus T_{f} is obtained from the cylinder X×[0, 1] by glueing the bottom to the top by the identification (x, 1) = (f(x), 0). There is a canonical map p: T_{f} → S^{1} which sends (x, t) to exp(2πit). It induces a canonical epimorphism π_{1}(T_{f}) → ℤ = π_{1}(S^{1}) if X is path-connected.

The following result is taken from [80, Theorem 6.63].

Theorem 3.10 (Vanishing of L^{2}-Betti numbers of mapping tori). Let f: X → X be a cellular selfmap of a connected CW -complex X and let π_{1}(T_{f}) ⟶^{ϕ} G ⟶^{ψ} ℤ be a factorization of the canonical epimorphism into epimorphisms ϕ and ψ. Suppose for given p ≥ 0 that b_{p}^{(2)}(G ×_{ϕ∘i} X̃; N (G)) < ∞ and b_{p−1}^{(2)}(G ×_{ϕ∘i} X̃; N (G)) < ∞ holds, where i: π_{1}(X) → π_{1}(T_{f}) is the map induced by the obvious inclusion of X into T_{f}. Let \overline{T_{f}} be the covering of T_{f} associated to ϕ, which is a free G-CW -complex. Then we get

b_{p}^{(2)}(\overline{T_{f}}; N (G)) = 0.

Proof. We give the proof in the special case where X is a connected finite CW - complex and ϕ = id, i.e. we show for a connected finite CW -complex X that b_{p}^{(2)}(T̃_{f}) = 0 for all p ≥ 0. For each positive integer d there is a finite d-sheeted covering \overline{T_{f}} → T_{f} associated to the subgroup of index d in π_{1}(T_{f}) which is the preimage of dℤ ⊆ ℤ under the canonical homomorphism π_{1}(T_{f}) → ℤ. There is a homotopy equivalence T_{f^{d}} → \overline{T_{f}}. We conclude from (3.2) and homotopy invariance of L^{2}-Betti numbers (see Theorem 2.7 (i))

b_{p}^{(2)}(T̃_{f}) = \frac{b_{p}^{(2)}(T̃_{f^{d}})}{d}.

There is a CW -complex structure on T_{f^{d}} with β_{p}(X)+β_{p−1}(X) p-cells, if β_{p}(X) is the number of p-cells in X. We conclude from Additivity of the dimension function (see Theorem 1.11 (iii)b)

b_{p}^{(2)}(T̃_{f^{d}}) ≤ dim_{N (π_{1}(T_{f^{d}}))} (N (π_{1}(T_{f^{d}})) ⊗_{ℤπ_{1}(T_{f^{d}})} C_{p}(T̃_{f^{d}}))

= β_{p}(X) + β_{p−1}(X).

This implies for all positive integers d

0 ≤ b_{p}^{(2)}(T̃_{f}) ≤ \frac{β_{p}(X) + β_{p−1}(X)}{d}.

Taking the limit for d → ∞ implies b_{p}^{(2)}(T̃_{f}) = 0.

3.8 Fibrations

The next result is proved in [80, Lemma 6.6. and Theorem 6.67]. The proof is based on standard spectral sequence arguments and the fact that the dimension function is defined for arbitrary N (G)-modules.

Theorem 3.11 (L^{2}-Betti numbers and fibrations).

(i) Let F ⟶^{i} E ⟶^{p} B be a fibration of connected CW -complexes. Consider a factorization p_{∗}: π_{1}(E) ⟶^{ϕ} G ⟶^{ψ} π_{1}(B) of the map induced by p into epimorphisms ϕ and ψ. Let i_{∗}: π_{1}(F) → π_{1}(E) be the homomorphism induced by the inclusion i. Suppose for a given integer d ≥ 1 that b_{p}^{(2)}(G ×_{ϕ∘i_{∗}} F̃; N (G)) = 0 for p ≤ d − 1 and b_{d}^{(2)}(G ×_{ϕ∘i_{∗}} F̃; N (G)) < ∞ holds. Suppose that π_{1}(B) contains an element of infinite order or finite subgroups of arbitrarily large order. Then b_{p}^{(2)}(G ×_{ϕ} Ẽ; N (G)) = 0 for p ≤ d;

(ii) Let F ⟶^{i} E → B be a fibration of connected CW -complexes. Consider a group homomorphism ϕ: π_{1}(E) → G. Let i_{∗}: π_{1}(F) → π_{1}(E) be the homomorphism induced by the inclusion i. Suppose that for a given integer d ≥ 0 the L^{2}-Betti number b_{p}^{(2)}(G ×_{ϕ∘i_{∗}} F̃; N (G)) vanishes for all p ≤ d. Then the L^{2}-Betti number b_{p}^{(2)}(G ×_{ϕ} Ẽ; N (G)) vanishes for all p ≤ d.

4 The Atiyah Conjecture

In this section we discuss the Atiyah Conjecture

Conjecture 4.1 (Atiyah Conjecture). Let G be a discrete group with an upper bound on the orders of its finite subgroups. Consider d ∈ ℤ, d ≥ 1 such that the order of every finite subgroup of G divides d. Let F be a field with ℚ ⊆ F ⊆ ℂ. The Atiyah Conjecture for (G, d, F) says that for any finitely presented F G-module M we have

d · dim_{N (G)} (N (G) ⊗_{F G} M) ∈ ℤ.

4.1 Reformulations of the Atiyah Conjecture

We present equivalent reformulations of the Atiyah Conjecture 4.1.

Theorem 4.2 (Reformulations of the Atiyah Conjecture). Let G be a discrete group. Suppose that there exists d ∈ ℤ, d ≥ 1 such that the order of every finite subgroup of G divides d. Let F be a field with ℚ ⊆ F ⊆ ℂ. Then the following assertions are equivalent:

(i) The Atiyah Conjecture 4.1 is true for (G, d, F), i.e. for every finitely presented F G-module M we have

d · dim_{N (G)} (N (G) ⊗_{F G} M) ∈ ℤ;

(ii) For every F G-module M we have

d · dim_{N (G)} (N (G) ⊗_{F G} M) ∈ ℤ ⨿ {∞}.

Proof. See [80, Lemma 10.7 and Remark 10.11].

We mention that the Atiyah Conjecture 4.1 is true for (G, d, F) if and only if for any finitely generated subgroup H ⊆ G the Atiyah Conjecture 4.1 is true for (H, d, F) (see [80, Lemma 10.4]).

The next result explains that the Atiyah Conjecture 4.1 for (G, d, ℚ) for a finitely generated group G is a statement about the possible values of L^{2}-Betti numbers.

Theorem 4.3 (Reformulations of the Atiyah Conjecture for F = ℚ). Let G be a finitely generated group with an upper bound d ∈ ℤ, d ≥ 1 on the orders of its finite subgroups. Then the following assertions are equivalent:

(i) The Atiyah Conjecture 4.1 is true for (G, d, ℚ);

(ii) For every free proper smooth cocompact G-manifold M without boundary

and p ∈ ℤ we have

d · b_{p}^{(2)}(M; N (G)) ∈ ℤ;

(iii) For every finite free G-CW -complex X and p ∈ ℤ we have

d · b_{p}^{(2)}(X; N (G)) ∈ ℤ;

(iv) For every G-space X and p ∈ ℤ we have

d · b_{p}^{(2)}(X; N (G)) ∈ ℤ ⨿ {∞}.

Proof. This follows from [80, Lemma 10.5] and Theorem 4.2.

We mention that all the explicit computations presented in Section 3 are compatible with the Atiyah Conjecture 4.1.

4.2 The Ring Theoretic Version of the Atiyah Conjecture

In this subsection we consider the following fundamental square of ring extensions

\begin{matrix}ℂG & ⟶^{i} & N (G) \\ ^{j}↓ & & ↓^{k} \\ D(G) & ⟶_{l} & U(G) \end{matrix}  (4.4)

which we explain next.

As before ℂG is the complex group ring and N (G) is the group von Neumann algebra.

By U(G) we denote the algebra of affiliated operators. Instead of its functional analytic definition we describe it algebraically, namely, it is the Ore localization of N (G) with respect to the multiplicative subset of non-trivial zerodivisors in N (G). The proof that this multiplicative subset satisfies the Ore condition and basic definitions and properties of Ore localization and of U(G) can be found for instance in [80, Sections 8.1 and 8.2]. In particular U(G) is flat when regarded as an N (G)-module. Moreover, the ring U(G) is a von Neumann regular ring, i.e. every finitely generated submodule of a projective module is a direct summand. This is a stronger condition than being semihereditary.

Given a finitely generated projective U(G)-module Q, there is a finitely generated projective N (G)-module P such that U(G) ⊗_{N (G)} P and Q are U(G)- isomorphic. If P_{0} and P_{1} are two finitely generated projective N (G)-modules, then P_{0} ≅_{N (G)} P_{1} ⇔ U(G) ⊗_{N (G)} P_{0} ≅_{U(G)} U(G) ⊗_{N (G)} P_{1}. This enables us to define a dimension function for dim_{U(G)} with properties analogous to dim_{N (G)} (see [80, Section 8.3], [98] or [99]).

Theorem 4.5. (Dimension function for arbitrary U(G)-modules). There exists precisely one dimension function

dim_{U(G)} : {U(G)-modules} → [0, ∞]

which satisfies:

(i) Extension Property

If M is an N (G)-module, then

dim_{U(G)} (U(G) ⊗_{N (G)} M) = dim_{N (G)}(M);

(ii) Additivity

If 0 → M_{0} → M_{1} → M_{2} → 0 is an exact sequence of U(G)-modules, then

dim_{U(G)}(M_{1}) = dim_{U(G)}(M_{0}) + dim_{U(G)}(M_{2});

(iii) Cofinality

Let {M_{i} | i ∈ I} be a cofinal system of submodules of M. Then

dim_{U(G)}(M) = sup{dim_{U(G)}(M_{i}) | i ∈ I};

(iv) Continuity

If K ⊆ M is a submodule of the finitely generated U(G)-module M, then

dim_{U(G)}(K) = dim_{U(G)}(\overline{K}).

Remark 4.6 (Comparing ℤ ⊆ ℚ and N (G) ⊆ U(G)). Recall the Slogan 1.16 that the group von Neumann algebra N (G) behaves like the ring of integers ℤ, provided one ignores the properties integral domain and Noetherian. This is supported by the construction and properties of U(G). Obviously U(G) plays the same role for N (G) as ℚ plays for ℤ as the definition of U(G) as the Ore localization of N (G) with respect to the multiplicative subset of non-zero-divisors and Theorem 4.5 show.

A subring R ⊆ S is called division closed if each element in R, which is invertible in S, is already invertible in R. It is called rationally closed if each square matrix over R, which is invertible over S, is already invertible over R. Notice that the intersection of division closed subrings of S is again division closed, and analogously for rationally closed subrings. Hence the following definition makes sense.

Definition 4.7 (Division and rational closure). Let S be a ring with subring R ⊆ S. The division closure D(R ⊆ S) or rational closure R(R ⊆ S) respectively is the smallest subring of S which contains R and is division closed or rationally closed respectively.

The ring D(G) appearing in the fundamental square (4.4) is the rational closure of ℂG in U(G).

Conjecture 4.8 (Ring theoretic version of the Atiyah Conjecture). Let G be a group for which there exists an upper bound on the orders of its finite subgroups. Then:

(R) The ring D(G) is semisimple;

(K) The composition

⨁_{H⊆G,|H|<∞} K_{0}(ℂH) ⟶^{a} K_{0}(ℂG) ⟶^{j} K_{0}(D(G))

is surjective, where a is induced by the various inclusions H → G.

Lemma 4.9. Let G be a group. Suppose that there exists d ∈ ℤ, d ≥ 1 such that the order of every finite subgroup of G divides d. If the group G satisfies the ring theoretic version of the Atiyah Conjecture 4.8, then the Atiyah Conjecture 4.1 for (G, d, ℂ) is true.

Proof. Let M be a finitely presented ℂG-module. Then D(G)⊗_{ℂG}M is a finitely generated projective D(G)-module since D(G) is semisimple by assumption. We obtain a well-defined homomorphism of abelian groups

D: K_{0}(D(G)) → ℝ, [P] ↦ dim_{U(G)} (U(G) ⊗_{D(G)} P ) .

Because of the fundamental square (4.4) and Theorem 4.5 (i) we have

dim_{N (G)}(N (G) ⊗_{ℂG} M) = D([D(G) ⊗_{ℂG} M]).

Hence it suffices to show that d · im(D) is contained in ℤ. Because of assumption (K) it suffices to check for each finite subgroup H ⊆ G and each finitely generated projective ℂH-module P

d · dim_{U(G)}(U(G) ⊗_{ℂG} ℂG ⊗_{ℂH} P) ∈ ℤ.

Example 1.14 and Theorem 1.18 imply

dim_{U(G)}(U(G) ⊗_{ℂG} ℂG ⊗_{ℂH} P) = dim_{N (G)}(N (G) ⊗_{ℂG} ℂG ⊗_{ℂH} P)
= dim_{N (G)}(N (G) ⊗_{N (H)} P)
= dim_{N (H)}(P)
= \frac{dim_{ℂ}(P)}{|H|}.

Obviously d · \frac{dim_{ℂ}(P)}{|H|} ∈ ℤ.

4.3 The Atiyah Conjecture for Torsion-Free Groups

Remark 4.10 (The Atiyah Conjecture in the torsion-free case). Let G be a torsion-free group. Then we can choose d = 1 in the Atiyah Conjecture 4.1. The Atiyah Conjecture 4.1 for (G, 1, F) says that dim_{N (G)}(N (G) ⊗_{F G} M) ∈ ℤ holds for every finitely presented F G-module M and Theorem 4.2 says that then this holds automatically for all F G-modules M with dim_{N (G)}(N (G) ⊗_{F G} M) < ∞. In the case, where F = ℚ and G is a torsion-free finitely generated group G, Theorem 4.3 implies that the Atiyah Conjecture 4.1 for (G, 1, F) is equivalent to the statement that b_{p}^{(2)}(X; N (G)) ∈ ℤ is true for all G-spaces X.

Remark 4.11 (The ring theoretic version of the Atiyah Conjecture in the torsion-free case). Let G be a torsion-free group. Then the ring theoretic version of the Atiyah Conjecture 4.8 reduces to the statement that D(G) is a skewfield. In this case we can assign to every D(G)-module N its dimension dim_{D(G)}(N) ∈ ℤ ⨿ {∞} in the usual way and we get for every ℂG-module M

dim_{N (G)}(N (G)⊗_{ℂG}M) = dim_{U(G)}(U(G)⊗_{ℂG}M) = dim_{D(G)}(D(G)⊗_{ℂG}M).

Example 4.12 (The case G = ℤ^{n}). In the case G = ℤ^{n} the fundamental square of ring extensions (4.4) can be identified with

ℂ[ℤ^{n}] ⟶ L^{∞}(T^{n})
↓  ↓
ℂ[ℤ^{n}]_{(0)} ⟶ MF (T^{n})

where MF (T^{n}) the ring of equivalence classes of measurable functions T^{n} → ℂ. We have already proved

dim_{N (ℤ^{n})}(N (ℤ^{n}) ⊗_{ℂ[ℤ^{n}]} M) = dim_{ℂ[ℤ^{n}]_{(0)}}(ℂ[ℤ^{n}]_{(0)} ⊗_{ℂ[ℤ^{n}]} M)

in Example 1.19.

4.4 The Atiyah Conjecture Implies the Kaplanski Conjecture

The following conjecture is a well-known conjecture about group rings.

Conjecture 4.13 (Kaplanski Conjecture). Let F be a field and let G be a torsion-free group. Then F G contains no non-trivial zero-divisors.

Theorem 4.14 (The Atiyah and the Kaplanski Conjecture). Let G be a torsion-free group and let F be a field with ℚ ⊆ F ⊆ ℂ. Then the Atiyah Conjecture 4.1 for (G, 1, F) implies the Kaplanski Conjecture 4.13 for F and G.

Proof. Let u ∈ F G be a zero-divisor. Then the kernel of the N (G)-map r_{u}: N (G) → N (G) given by right multiplication with u is non-trivial. Since N (G) is semihereditary, the image of r_{u} is projective. Hence both ker(r_{u}) and N (G)/ ker(r_{u}) are finitely generated projective N (G)-modules. Additivity of dim_{N (G)} implies

0 < dim_{N (G)}(ker(r_{u})) ≤ dim_{N (G)}(N (G)) = 1.

We conclude from Remark 4.10 that dim_{N (G)}(ker(r_{u})) is an integer. Additivity of dim_{N (G)} implies

dim_{N (G)} (N (G)/ ker(r_{u})) = 0.

We conclude N (G)/ ker(r_{u}) = 0 and hence u = 0.

4.5 The Status of the Atiyah Conjecture

Let l^{∞}(G, ℝ) be the space of equivalence classes of bounded functions from G to ℝ with the supremum norm. Denote by 1 the constant function with value 1.

Definition 4.15 (Amenable group). A group G is called amenable, if there is a (left) G-invariant linear operator μ: l^{∞}(G, ℝ) → ℝ with μ(1) = 1, which satisfies for all f ∈ l^{∞}(G, ℝ)

inf{f(g) | g ∈ G} ≤ μ(f) ≤ sup{f(g) | g ∈ G}.

The latter condition is equivalent to the condition that μ is bounded and μ(f) ≥ 0 if f(g) ≥ 0 for all g ∈ G.

Definition 4.16 (Elementary amenable group). The class of elementary amenable groups EAM is defined as the smallest class of groups which has the following properties:

(i) It contains all finite and all abelian groups;

(ii) It is closed under taking subgroups;

(iii) It is closed under taking quotient groups;

(iv) It is closed under extensions, i.e. if 1 → H → G → K → 1 is an exact sequence of groups and H and K belong to EAM, then also G ∈ EAM;

(v) It is closed under directed unions, i.e. if {G_{i} | i ∈ I} is a directed system of subgroups such that G = ⋃_{i∈I} G_{i} and each G_{i} belongs to EAM, then G ∈ EAM. (Directed means that for two indices i and j there is a third index k with G_{i}, G_{j} ⊆ G_{k}.)

The class of amenable groups satisfies all the conditions appearing in Definition 4.16. Hence every elementary amenable group is amenable. The converse is not true.

Definition 4.17 (Linnell’s class of groups C). Let C be the smallest class of groups, which contains all free groups and is closed under directed unions and extensions with elementary amenable quotients.

The next result is due to Linnell [65].

Theorem 4.18 (Linnell’s Theorem). Let G be a group in C. Suppose that there exists d ∈ ℤ, d ≥ 1 such that the order of every finite subgroup of G divides d. Then the ring theoretic version of the Atiyah Conjecture 4.8 for G and hence the Atiyah Conjecture 4.1 for (G, d, ℂ) are true.

The next definition and the next theorem are due to Schick [101].

Definition 4.19. Let D be the smallest non-empty class of groups such that (i) If p: G → A is an epimorphism of a torsion-free group G onto an elemen-

tary amenable group A and if p^{−1}(B) ∈ D for every finite group B ⊆ A, then G ∈ D;

(ii) D is closed under taking subgroups;

(iii) D is closed under colimits and inverse limits over directed systems.

Theorem 4.20. (i) If the group G belongs to D, then G is torsion-free and the Atiyah Conjecture 4.1 for (G, 1, ℚ) is true for G;

(ii) The class D is closed under direct sums, direct products and free products. Every residually torsion-free elementary amenable group belongs to D.

More information about the status of the Atiyah Conjecture 4.1 can be found for instance in [80, Subsection 10.1.3].

4.6 Groups Without Bound on the Order of Its Finite Subgroups

Given a group G, let FIN (G) be the set of finite subgroups of G. Denote by

\frac{1}{|FIN (G)|}ℤ ⊆ ℚ  (4.21)

the additive subgroup of ℝ generated by the set of rational numbers \{\frac{1}{|H|} | H ∈ FIN (G)}.

There is the following formulation of the Atiyah Conjecture for arbitrary groups in the literature.

Conjecture 4.22 (Atiyah Conjecture for arbitrary groups G). A group G satisfies the Atiyah Conjecture if for every finitely presented ℂG-module M we have

dim_{N (G)}(N (G) ⊗_{ℂG} M) ∈ \frac{1}{|FIN (G)|}ℤ.

There do exist counterexamples to this conjecture. The lamplighter group L is defined by the semidirect product

L := (⨁_{n∈ℤ} ℤ/2) ⋊ ℤ

with respect to the shift automorphism of ⨁_{n∈ℤ} ℤ/2, which sends (x_{n})_{n∈ℤ} to (x_{n−1})_{n∈ℤ}. Let e_{0} ∈ ⨁_{n∈ℤ} ℤ/2 be the element whose entries are all zero except the entry at 0. Denote by t ∈ ℤ the standard generator of ℤ which we will also view as an element of L. Then {e_{0}t, t} is a set of generators for L. The associated Markov operator M: l^{2}(G) → l^{2}(G) is given by right multiplication with \frac{1}{4} ·(e_{0}t+t+(e_{0}t)^{−1}+t^{−1}). It is related to the Laplace operator Δ_{0} : l^{2}(G) → l^{2}(G) of the Cayley graph of G by Δ_{0} = 4 · id −4 · M. The following result is a special case of the main result in the paper of Grigorchuk and Żuk [41, Theorem 1 and Corollary 3] (see also [40]). An elementary proof can be found in [22].

Theorem 4.23 (Counterexample to the Atiyah Conjecture for arbitrary groups). The von Neumann dimension of the kernel of the Markov operator M of the lamplighter group L associated to the set of generators {e_{0}t, t} is 1/3. In particular L does not satisfy the Atiyah Conjecture 4.22.

To the author’s knowledge there is no example of a group G for which there is a finitely presented ℂG-module M such that dim_{N (G)}(N (G) ⊗_{ℤG} M) is irrational.

Let A = ⨁_{n∈ℤ} ℤ/2. Because this group is locally finite, it satisfies the Atiyah Conjecture for arbitrary groups 4.22, i.e. dim_{N (G)}(N (G) ⊗_{ℂA} M) ∈ ℤ[1/2] for every finitely presented ℂA-module M. On the other hand, each nonnegative real number r can be realized as dim_{N (G)}(N (G) ⊗_{ℂA} M) for a finitely generated ℂA-module (see [80, Example 10.13]). Notice that there is no upper bound on the orders of finite subgroups of A, so that this is no contradiction to Theorem 4.2.

5 Flatness Properties of the Group von Neumann Algebra

The proof of next result can be found in [77, Theorem 5.1] or [80, Theorem 6.37].

Theorem 5.1. (Dimension-flatness of N (G) over ℂG for amenable G). Let G be amenable and M be a ℂG-module. Then

dim_{N (G)} (Tor_{p}^{ℂG}(N (G), M)) = 0  for p ≥ 1,

where we consider N (G) as an N (G)-ℂG-bimodule.

It implies using an easy spectral sequence argument

Theorem 5.2 (L^{2}-Betti numbers and homology in the amenable case). Let G be an amenable group and X be a G-space. Then

(i) b_{p}^{(2)}(X; N (G)) = dim_{N (G)} (N (G) ⊗_{ℂG} H_{p}^{sing}(X; ℂ));

(ii) Suppose that X is a G-CW -complex with m(X; G) < ∞. Then

χ^{(2)}(X) = ∑_{c∈I(X)}(−1)^{dim(c)} · |G_{c}|^{−1}
= ∑_{p≥0}(−1)^{p} · dim_{N (G)} (N (G) ⊗_{ℂG} H_{p}(X; ℂ)) .

Further applications of Theorem 5.1 will be discussed in Section 6 and Section 7.

Conjecture 5.3. (Amenability and dimension-flatness of N (G) over ℂG). A group G is amenable if and only if for every ℂG-module M

dim_{N (G)} (Tor_{p}^{ℂG}(N (G), M)) = 0  for p ≥ 1

holds.

Remark 5.4 (Evidence for Conjecture 5.3). Theorem 5.1 proves the “only if”-statement of Conjecture 5.3. Some evidence for the “if”-statement of Conjecture 5.3 comes from the following fact. Notice that a group which contains a non-abelian free group as a subgroup, cannot be amenable.

Suppose that G contains a free group ℤ ∗ ℤ of rank 2 as a subgroup. Notice that S^{1} ∨ S^{1} is a model for B(ℤ ∗ ℤ). Its cellular ℂ[ℤ ∗ ℤ]-chain complex yields an exact sequence 0 → ℂ[ℤ∗ℤ]^{2} → ℂ[ℤ∗ℤ] → ℂ → 0, where ℂ is equipped with the trivial ℤ ∗ ℤ-action. One easily checks b_{1}^{(2)}({ \atop S^{1} ∨ S^{1}}) = −χ(S^{1} ∨ S^{1}) = 1. This implies

dim_{N (ℤ∗ℤ)} (Tor_{1}^{ℂ[ℤ∗ℤ]}(N (ℤ ∗ ℤ), ℂ)) = 1.

We conclude from Theorem 1.18 (i)

N (G) ⊗_{N (ℤ∗ℤ)} Tor_{1}^{ℂ[ℤ∗ℤ]}(N (ℤ ∗ ℤ), ℂ) = Tor_{1}^{ℂG}(N (G), ℂG ⊗_{ℂ[ℤ∗ℤ]} ℂ).

Theorem 1.18 (ii) implies

dim_{N (G)} (Tor_{1}^{ℂG}(N (G), ℂG ⊗_{ℂ[ℤ∗ℤ]} ℂ)) = 1.

One may ask for which groups the von Neumann algebra N (G) is flat as a ℂG-module. This is true if G is virtually cyclic, i.e. G is finite or contains ℤ as a normal subgroup of finite index. There is some evidence for the following conjecture (see [77, Remark 5.15]).

Conjecture 5.5 (Flatness of N (G) over ℂG). The group von Neumann algebra N (G) is flat over ℂG if and only if G is virtually cyclic.

6 Applications to Group Theory

Recall the Definition 2.1 of the L^{2}-Betti numbers of a group G by b_{p}^{(2)}(G) := b_{p}^{(2)}(EG; N (G)). In this section we present tools for and examples of computations of the L^{2}-Betti numbers and discuss applications to group theory. We will explain in Remark 7.8 that for a torsion-free group with a model of finite type for BG the knowledge of b_{p}^{(2)}(G; N (G)) is the same as the knowledge of the reduced L^{2}-homology H_{p}^{(2)}(EG, l^{2}(G)), or, equivalently, of PH_{p}^{G}(EG; N (G)) if G satisfies the Atiyah Conjecture 4.1 for (G, 1, ℚ).

6.1 L^{2}-Betti Numbers of Groups

Theorem 2.7 implies:

Theorem 6.1 (L^{2}-Betti numbers and Betti numbers of groups). In the sequel we use the conventions 0·∞ = 0, r·∞ = ∞ for r ∈ (0, ∞] and r+∞ = ∞ for r ∈ [0, ∞] and put |G|^{−1} = 0 for |G| = ∞. Let G_{1}, G_{2}, … be a sequence of non-trivial groups.

(i) Free amalgamated products For r ∈ {2, 3, …} ⨿ {∞} we get

b_{0}^{(2)}(∗_{i=1}^{r}G_{i}) = 0;
b_{1}^{(2)}(∗_{i=1}^{r}G_{i}) = \{ \begin{matrix}r − 1 + ∑_{i=1}^{r} (b_{1}^{(2)}(G_{i}) − \frac{1}{|G_{i}|}) & , if r < ∞; \\ ∞ & , if r = ∞; \end{matrix}

b_{p}^{(2)}(∗_{i=1}^{r}G_{i}) = ∑_{i=1}^{r} b_{p}^{(2)}(G_{i})  for p ≥ 2;

b_{p}(∗_{i=1}^{r}G_{i}) = ∑_{i=1}^{r} b_{p}(G_{i})  for p ≥ 1;

(ii) Künneth formula

b_{p}^{(2)}(G_{1} × G_{2}) = ∑_{i=0}^{p} b_{i}^{(2)}(G_{1}) · b_{p−i}^{(2)}(G_{2});

b_{p}(G_{1} × G_{2}) = ∑_{i=0}^{p} b_{i}(G_{1}) · b_{p−i}(G_{2});

(iii) Restriction to subgroups of finite index For a subgroup H ⊆ G of finite index [G : H] we get

b_{p}^{(2)}(H) = [G : H] · b_{p}^{(2)}(G);

(iv) Extensions with finite kernel Let 1 → H → G → Q → 1 be an extension of groups with finite H. Then

b_{p}^{(2)}(Q) = |H| · b_{p}^{(2)}(G);

(v) Zero-th L^{2}-Betti number

We have b_{0}^{(2)}(G) = 0 for |G| = ∞ and b_{0}^{(2)}(G) = |G|^{−1} for |G| < ∞.

Example 6.2 (Independence of L^{2}-Betti numbers and Betti numbers). Given an integer l ≥ 1 and a sequence r_{1}, r_{2}, …, r_{l} of non-negative rational numbers, we can construct a group G such that BG is of finite type and

b_{p}^{(2)}(G) = \{ \begin{matrix}r_{p} & for 1 ≤ p ≤ l; \\ 0 & for l + 1 ≤ p; \end{matrix}
b_{p}(G) = 0  for p ≥ 1,

holds as follows.

For integers m ≥ 0, n ≥ 1 and i ≥ 1 define

G_{i}(m, n) = ℤ/n × (∗_{k=1}^{2m+2}ℤ/2) × (∏_{j=1}^{i−1} ∗_{l=1}^{4}ℤ/2)

One easily checks using Theorem 6.1.

b_{i}^{(2)}(G_{i}(m, n)) = \frac{m}{n};
b_{p}^{(2)}(G_{i}(m, n)) = 0  for p ≠ i;
b_{p}(G_{i}(m, n)) = 0  for p ≥ 1.

Define the desired group G as follows. For l = 1 put G = G_{1}(m, n) if r_{1} = m/n. It remains to treat the case l ≥ 2. Choose integers n ≥ 1 and k ≥ l with r_{1} = \frac{k−2}{n}. Fix for i = 2, 3, …, k integers m_{i} ≥ 0 and n_{i} ≥ 1 such that \frac{m_{i}}{n·n_{i}} = r_{i} holds for 1 ≤ i ≤ l and m_{i} = 0 holds for i > l. Put

G = ℤ/n × ∗_{i=2}^{k}G_{i}(m_{i}, n_{i}).

One easily checks using Theorem 6.1 that G has the prescribed L^{2}-Betti numbers and Betti numbers and a model for BG of finite type.

On the other hand we can construct for any sequence n_{1}, n_{2}, … of nonnegative integers a CW -complex X of finite type such that b_{p}(X) = n_{p} and b_{p}^{(2)}(X̃) = 0 holds for p ≥ 1, namely take

X = B(ℤ/2 ∗ ℤ/2) × ⋁_{p=1}^{∞} (⋁_{i=1}^{n_{p}} S^{p}) .

This example shows by considering the (l + 1)-skeleton that for a finite connected CW -complex X the only general relation between the L^{2}-Betti numbers b_{p}^{(2)}(X̃) of its universal covering X̃ and the Betti numbers b_{p}(X) of X is given by the Euler-Poincaré formula (see Theorem 2.7 (x))

∑_{p≥0}(−1)^{p} · b_{p}^{(2)}(X̃) = χ(X) = ∑_{p≥0}(−1)^{p} · b_{p}(X).

6.2 Vanishing of L^{2}-Betti Numbers of Groups

Let d be a non-negative integer or d = ∞. In this subsection we want to investigate the class of groups

B_{d} := {G | b_{p}^{(2)}(G) = 0 for 0 ≤ p ≤ d}.  (6.3)

Notice that B_{0} is the class of infinite groups by Theorem 6.1 (v).

Theorem 6.4. Let d be a non-negative integer or d = ∞. Then:

(i) The class B_{∞} contains all infinite amenable groups;

(ii) If G contains a normal subgroup H with H ∈ B_{d}, then G ∈ B_{d};

(iii) If G is the union of a directed system of subgroups {G_{i} | i ∈ I} such that each G_{i} belongs to B_{d}, then G ∈ B_{d};

(iv) Suppose that there are groups G_{1} and G_{2} and group homomorphisms ϕ_{i}: G_{0} → G_{i} for i = 1, 2 such that ϕ_{1} and ϕ_{2} are injective, G_{0} belongs to B_{d−1}, G_{1} and G_{2} belong to B_{d} and G is the amalgamated product G_{1}∗_{G_{0}} G_{2} with respect to ϕ_{1} and ϕ_{2}. Then G belongs to B_{d};

(v) Let 1 → H → G → K → 1 be an exact sequence of groups such that b_{p}^{(2)}(H) is finite for all p ≤ d. Suppose that K is infinite amenable or suppose that BK has finite d-skeleton and there is an injective endomorphism j: K → K whose image has finite index, but is not equal to K. Then G ∈ B_{d};

(vi) Let 1 → H → G → K → 1 be an exact sequence of groups such that H ∈ B_{d−1}, b_{d}^{(2)}(H) < ∞ and K contains an element of infinite order or finite subgroups of arbitrary large order. Then G ∈ B_{d};

(vii) Let 1 → H → G → K → 1 be an exact sequence of infinite countable groups such that b_{1}^{(2)}(H) < ∞. Then G ∈ B_{1}.

Proof. (i) We get b_{p}^{(2)}(G) = 0 for p = 0 from Theorem 6.1 (v). The case p ≥ 1 follows from Theorem 5.2 (i) since H_{p}^{sing}(EG; ℂ) = 0 for p ≥ 1. (ii) Apply Theorem 3.11 (ii) to the fibration BH → BG → B(G/H). (iii) The proof is based on a colimit argument. See [80, Theorem 7.2 (3)]. (iv) The proof is based on a Mayer-Vietoris argument. See [80, Theorem 7.2 (4)].

(v) See [80, Theorem 7.2 (5)]. (vi) This follows from Theorem 3.11 (i) applied to the fibration BH → BG → B(G/H). (vii) This is proved by Gaboriau [38, Theorem 6.8].

More information about the vanishing of the first L^{2}-Betti number can be found for instance in [4]. Obviously the following is true Lemma 6.5. If G belongs to B_{∞}, then χ^{(2)}(G) = 0.

Remark 6.6 (The Theorem of Cheeger and Gromov). We rediscover from Theorem 6.4 the result of Cheeger and Gromov [15] that all the L^{2}-Betti numbers of an infinite amenable group G vanish. A detailed comparison of our approach and the approach by Cheeger and Gromov to L^{2}-Betti numbers can be found in [80, Remark 6.76].

Remark 6.7 (Advantage of the general definition of L^{2}-Betti numbers). Recall that we have given criterions for G ∈ B_{∞} in Theorem 6.4. Now it becomes clear why it is worth while to extend the classical notion of the Euler characteristic χ(G) := χ(BG) for groups G with finite BG to arbitrary groups. For instance it may very well happen for a group G with finite BG that G contains a normal group H which is not even finitely generated and has in particular no finite model for BH and which belongs to B_{∞} (for instance, H is amenable). Then the classical Euler characteristic is not defined any more for H, but we can still conclude that the classical Euler characteristic of G vanishes by Remark 2.19, Theorem 6.4 and Lemma 6.5.

6.3 L^{2}-Betti Numbers of Some Specific Groups

Example 6.8 (Thompson’s group). Next we explain the following observation about Thompson’s group F. It is the group of orientation preserving dyadic PL-automorphisms of [0, 1], where dyadic means that all slopes are integral powers of 2 and the break points are contained in ℤ[1/2]. It has the presentation

F = ⟨x_{0}, x_{1}, x_{2}, … | x_{i}^{−1}x_{n}x_{i} = x_{n+1} for i < n⟩.

This group has some very interesting properties. Its classifying space BF is of finite type [8] but is not homotopy equivalent to a finite dimensional CW - complex since F contains ℤ^{n} as a subgroup for all n ≥ 0 [8, Proposition 1.8]. It is not elementary amenable and does not contain a subgroup which is free on two generators [7], [10]. Hence it is a very interesting question whether F is amenable or not. We conclude from Theorem 6.4 (i) that a necessary condition for F to be amenable is that b_{p}^{(2)}(F) vanishes for all p ≥ 0. By [80, Theorem 7.10] this condition is satisfied.

Example 6.9 (Artin groups). Davis and Leary [19] compute for every Artin group A the reduced L^{2}-cohomology and thus the L^{2}-Betti numbers of the universal covering S̃_{A} of its Salvetti complex S_{A}. The Salvetti complex S_{A} is a CW-complex which is conjectured to be a model for the classifying space BA of A. This conjecture is known to be true in many cases and implies that the L^{2}-Betti numbers of A are given by the L^{2}-Betti numbers of S̃_{A}.

Example 6.10 (Right angled Coxeter groups). The L^{2}-homology and the L^{2}-Betti numbers of right angled Coxeter groups are treated by Davis and Okun [20]. More details will be given in Remark 9.6.

Example 6.11 (Fundamental groups of surfaces and 3-manifolds). Let G be the fundamental group of a compact orientable surface F_{g}^{d} of genus g with d boundary components. Suppose that G is non-trivial which is equivalent to the condition that d ≥ 1 or g ≥ 1. Then F_{g}^{d} is a model for BG and we have

computed b_{p}^{(2)}(G) = b_{p}^{(2)}(F̃_{g}^{d}) in Subsection 3.3.

Let G be the fundamental group of a compact orientable 3-manifold M. The case |G| < ∞ is clear, since then the universal covering is homotopy equivalent to a sphere or contractible. So let us assume |G| = ∞. Under the condition that M in non-exceptional, we have computed b_{p}^{(2)}(M̃) in Theorem 3.4. If M is prime, then either M = S^{1} × S^{2} and G = ℤ and b_{p}^{(2)}(G) = 0 for all p ≥ 0 or M is irreducible, in which case M is aspherical and b_{p}^{(2)}(G) = b_{p}^{(2)}(M̃).

Suppose that M is not prime. Then still b_{1}^{(2)}(G) = b_{1}^{(2)}(M̃) by Theorem 2.7 (i)a since the classifying map M → BG is 2-connected. Suppose the prime decomposition of M looks like M = #_{i=1}^{r}M_{i}. Then G = ∗_{i=1}^{r}G_{i} for G_{i} = π_{1}(M_{i}). We know b_{p}^{(2)}(G_{i}) for each i if each M_{i} is non-exceptional and we get b_{p}^{(2)}(G) = ∑_{i=1}^{r} b_{p}^{(2)}(G_{i}) for p ≥ 2 from Theorem 6.1 (i).

Example 6.12 (One relator groups). Let G = ⟨g_{1}, g_{2}, … g_{s} | R⟩ be a torsion-free one relator group for s ∈ {2, 3 …}⨿{∞} and one non-trivial relation R. Then

b_{p}^{(2)}(G) = \{ \begin{matrix}0 & if p ≠ 1; \\ s − 2 & if p = 1 and s < ∞; \\ ∞ & if p = 1 and s = ∞. \end{matrix}

We only treat the case s < ∞, the general case is obtained from it by taking the free amalgamated product with a free group. Because the 2-dimensional CW -complex X associated to the given presentation is a model for BG (see [85, chapter III §§9 -11]) and satisfies χ(X) = s − 2, it suffices to prove b_{2}^{(2)}(G) = 0. We sketch the argument of Dicks and Linnell for this claim. Howie [56] has shown that such a group G is locally indicable and hence left-orderable. A result of Linnell [64, Theorem 2] for left-orderable groups says that an element α ∈ ℂG with α ≠ 0 is a non-zero-divisor in U(G). This implies that the second differential c_{2}^{U(G)} in the chain complex U(G) ⊗_{ℂG} C_{∗}(EG) is injective. Since U(G) is flat over N (G), we get from Theorem 4.5

b_{2}^{(2)}(G) = dim_{N (G)} (H_{p}^{G}(EG; N (G))) = dim_{U(G)} (H_{p}^{G}(EG; U(G)))

= dim_{U(G)} (ker (c_{2}^{U(G)})) = 0.

Linnell has an extensions of this argument to non-torsion-free one-relator groups G with s ≥ 2 generators. (The case s = 1 is obvious.) Such a group contains a cyclic subgroup ℤ/k such that any finite subgroup is subconjugated to ℤ/k and then

b_{p}^{(2)}(G) = \{ \begin{matrix}0 & if p ≠ 1; \\ s − 1 − \frac{1}{k} & if p = 1 and s < ∞; \\ ∞ & if p = 1 and s = ∞. \end{matrix}

Example 6.13 (Lattices). Let L be a connected semisimple Lie group with finite center such that its Lie algebra has no compact ideal. Let G ⊆ L be a lattice, i.e. a discrete subgroup of finite covolume. We want to compute its L^{2}-Betti numbers. There is a subgroup G_{0} ⊆ G of finite index which is torsionfree. Since b_{p}^{(2)}(G) = [G : G_{0}] · b_{p}^{(2)}(G_{0}), it suffices to treat the case G = G_{0}, i.e. G ⊆ L is a torsion-free lattice.

Let K ⊆ L be a maximal compact subgroup. Put M = G\L/K. Then the space L/K = M̃ is a symmetric space of non-compact type. We have already mentioned in Theorem 3.5 that the work of Borel [6] implies for cocompact G that b_{p}^{(2)}(G) = b_{p}^{(2)}(M̃) ≠ 0 if and only if f-rk(M̃) = 0 and 2p = dim(M). This is actually true without the condition “cocompact”, because the condition “finite covolume” is enough.

Next we deal with the general case of a connected Lie group L. Let Rad(L) be its radical. One can choose a compact normal subgroup K ⊆ L such that R = Rad(L) × K is a normal subgroup of L and the quotient L_{1} = L/R is a semisimple Lie group such that its Lie algebra has no compact ideal. Then G_{1} = L/L ∩ R is a lattice in L_{1} and G ∩ R is a lattice in R. The group G ∩ R is a normal amenable subgroup of G. If G ∩ R is infinite, we get b_{p}^{(2)}(G) = 0 for all p ≥ 0 from Theorem 6.4. If G ∩ R is finite, we get b_{p}^{(2)}(G) = |G ∩ R|^{−1} · b_{p}^{(2)}(G_{1}) for all p ≥ 0 from Theorem 6.1 (iv). If the center of L_{1} is infinite, the center of G_{1} must also be infinite and hence b_{p}^{(2)}(G_{1}) = 0 for all p ≥ 0 by Theorem 6.4. Suppose that the center of L_{1} is finite. Then we know already how to compute the L^{2}-Betti numbers of G_{1} from the explanation above.

Given a lattice G in a connected Lie group, b_{1}^{(2)}(G) > 0 is true if and only if G is commensurable with a torsion-free lattice in P SL_{2}(ℝ), or, equivalently commensurable with a surface group for genus ≥ 2 or a finitely generated nonabelian free group (see Eckmann [27] or Lott [68, Theorem 2]).

6.4 Deficiency and L^{2}-Betti Numbers of Groups

Let G be a finitely presented group. Define its deficiency def(G) to be the maximum g(P) − r(P), where P runs over all presentations P of G and g(P) is the number of generators and r(P) is the number of relations of a presentation P.

Next we reprove the well-known fact that the maximum appearing in the definition of the deficiency does exist.

Lemma 6.14. Let G be a group with finite presentation

P = ⟨s_{1}, s_{2}, …, s_{g} | R_{1}, R_{2}, …, R_{r}⟩

Let ϕ: G → K be any group homomorphism. Then

g(P) − r(P) ≤ 1 − b_{0}^{(2)}(K ×_{ϕ} EG; N (K)) + b_{1}^{(2)}(K ×_{ϕ} EG; N (K))

− b_{2}^{(2)}(K ×_{ϕ} EG; N (K)).

Proof. Given a presentation P with g generators and r relations, let X be the associated finite 2-dimensional CW -complex. It has one 0-cell, g 1-cells, one for each generator, and r 2-cells, one for each relation. There is an obvious isomorphism from π_{1}(X) to G so that we can choose a map f: X → BG which induces an isomorphism on the fundamental groups. It induces a 2-connected K-equivariant map \overline{f}: K ×_{ϕ} X̃ → K ×_{ϕ} ẼG. We conclude from Theorem 2.7 (i)a

b_{p}^{(2)}(K ×_{ϕ} X̃; N (K)) = b_{p}^{(2)}(K ×_{ϕ} EG; N (K)) for p = 0, 1;
b_{2}^{(2)}(K ×_{ϕ} X̃; N (K)) ≥ b_{2}^{(2)}(K ×_{ϕ} EG; N (K)).

We conclude from the L^{2}-Euler-Poincaré formula (see Theorem 2.18 (i))

g − r = 1 − χ^{(2)}(K ×_{ϕ} X̃; N (K))
= 1 − b_{0}^{(2)}(K ×_{ϕ} X̃; N (K)) + b_{1}^{(2)}(K ×_{ϕ} X̃; N (K))
−b_{2}^{(2)}(K ×_{ϕ} X̃; N (K))
≤ 1 − b_{0}^{(2)}(K ×_{ϕ} EG; N (K)) + b_{1}^{(2)}(K ×_{ϕ} EG; N (K))
−b_{2}^{(2)}(K ×_{ϕ} EG; N (K)).

Example 6.15 (Deficiency of some groups). Sometimes the deficiency is realized by the “obvious” presentation. For instance the deficiency of a free group ⟨s_{1}, s_{2}, …, s_{g} | ∅⟩ on g letters is indeed g. The cyclic group ℤ/n of order n has the presentation ⟨t | t^{n} = 1⟩ and its deficiency is 0. The group ℤ/n × ℤ/n has the presentation ⟨s, t | s^{n}, t^{n}, [s, t]⟩ and its deficiency is −1.

Remark 6.16 (Non-additivity of the deficiency). The deficiency is not additive under free products by the following example which is a special case of a more general example due to Hog, Lustig and Metzler [55, Theorem 3 on page 162]. The group (ℤ/2 × ℤ/2) ∗ (ℤ/3 × ℤ/3) has the obvious presentation

⟨s_{0}, t_{0}, s_{1}, t_{1} | s_{0}^{2} = t_{0}^{2} = [s_{0}, t_{0}] = s_{1}^{3} = t_{1}^{3} = [s_{1}, t_{1}] = 1⟩

One may think that its deficiency is −2. However, it turns out that its deficiency is −1, realized by the following presentation

⟨s_{0}, t_{0}, s_{1}, t_{1} | s_{0}^{2} = 1, [s_{0}, t_{0}] = t_{0}^{2}, s_{1}^{3} = 1, [s_{1}, t_{1}] = t_{1}^{3}, t_{0}^{2} = t_{1}^{3}⟩.

This shows that it is important to get upper bounds on the deficiency of groups. Writing down presentations gives lower bounds, but it is not clear whether a given presentation realizes the deficiency.

Lemma 6.17. Let G be a finitely presented group and let ϕ: G → K be a homomorphism such that b_{1}^{(2)}(K ×_{ϕ} EG; N (K)) = 0. Then

(i) def(G) ≤ 1;

(ii) Let M be a closed oriented 4-manifold with G as fundamental group. Then

| sign(M)| ≤ χ(M).

Proof. (i) This follows directly from Lemma 6.14. (ii) This is a consequence of the L^{2}-Signature Theorem due to Atiyah [2]. Details of the proof can be found in [80, Lemma 7.22].

Theorem 6.18. Let 1 → H ⟶^{i} G ⟶^{q} K → 1 be an exact sequence of infinite groups. Suppose that G is finitely presented and one of the following conditions is satisfied:

(i) b_{1}^{(2)}(H) < ∞;

(ii) The classical first Betti number of H satisfies b_{1}(H) < ∞ and K belongs to B_{1}.

Then

(i) def(G) ≤ 1;

(ii) Let M be a closed oriented 4-manifold with G as fundamental group. Then

| sign(M)| ≤ χ(M).

Proof. If condition (i) is satisfied, then b_{p}^{(2)}(G) = 0 for p = 0, 1 by Theorem 6.4 (vii), and the claim follows from Lemma 6.17.

Suppose that condition (ii) is satisfied. There is a spectral sequence converging to H_{p+q}^{K}(K ×_{q} EG; N (K)) with E^{2}-term

E_{p,q}^{2} = Tor_{p}^{ℂK}(H_{q}(BH; ℂ), N (K))

[108, Theorem 5.6.4 on page 143]. Since H_{q}(BH; ℂ) is ℂ with the trivial Kaction for q = 0 and finite dimensional as complex vector space by assumption for q = 1, we conclude dim_{N (K)}(E_{p,q}^{2}) = 0 for p + q = 1 from the assumption b_{1}^{(2)}(K) = 0. This implies b_{1}^{(2)}(K ×_{q} EG; N (K)) = 0 and the claim follows from Lemma 6.17.

Theorem 6.18 generalizes results in [29], [58], where also some additional information is given. Furthermore see [49], [63]. We mention the result of Hitchin [54] that a connected closed oriented smooth 4-manifold which admits an Einstein metric satisfies the stronger inequality | sign(M)| ≤ \frac{2}{3} · χ(M).

Finally we mention the following result of Lott [68, Theorem 2] (see also [28]) which generalizes a result of Lubotzky [70]. The statement we present here is a slight improvement of Lott’s result due to Hillman [53].

Theorem 6.19 (Lattices of positive deficiency). Let L be a connected Lie group. Let G be a lattice in L. If def(G) > 0, then one of the following assertions holds:

(i) G is a lattice in P SL_{2}(ℂ);

(ii) def(G) = 1. Moreover, either G is isomorphic to a torsion-free nonuniform lattice in ℝ × P SL_{2}(ℝ) or P SL_{2}(ℂ), or G is ℤ or ℤ^{2}.

7 G- and K-Theory

In this section we discuss the projective class group K_{0}(N (G)) of a group von Neumann algebra. We present applications of its computation to G_{0}(ℂG) and the Whitehead group Wh(G) of a group G.

7.1 The K_{0}-group of a Group von Neumann Algebra

In this subsection we want to investigate the projective class group of a group von Neumann algebra.

Definition 7.1 (Definition of K_{0}(R) and G_{0}(R)). Let R be an (associative) ring (with unit). Define its projective class group K_{0}(R) to be the abelian group whose generators are isomorphism classes [P] of finitely generated projective Rmodules P and whose relations are [P_{0}] + [P_{2}] = [P_{1}] for any exact sequence 0 → P_{0} → P_{1} → P_{2} → 0 of finitely generated projective R-modules.

Define the Grothendieck group of finitely generated modules G_{0}(R) analogously but replace finitely generated projective with finitely generated.

The group K_{0} is known for any von Neumann algebra (see for instance [80, Subsection 9.2.1]. For simplicity we only treat the von Neumann algebra N (G) of a group here.

The next result is taken from [60, Theorem 7.1.12 on page 462, Proposition 7.4.5 on page 483, Theorem 8.2.8 on page 517, Proposition 8.3.10 on page 525, Theorem 8.4.3 on page 532].

Theorem 7.2 (The universal trace). There is a map

tr_{N (G)}^{u} : N (G) → Z(N (G))

into the center Z(N (G)) of N (G) called the center valued trace or universal trace of N (G), which is uniquely determined by the following two properties:

(i) tr_{N (G)}^{u} is a trace with values in the center, i.e. tr_{N (G)}^{u} is ℂ-linear, for a ∈ N (G) with a ≥ 0 we have tr_{N (G)}^{u}(a) ≥ 0 and tr_{N (G)}^{u}(ab) = tr_{N (G)}^{u}(ba) for all a, b ∈ N (G); (ii) tr_{N (G)}^{u}(a) = a for all a ∈ Z(N (G)).

The map tr_{N (G)}^{u} has the following further properties:

(iii) tr_{N (G)}^{u} is faithful, i.e. tr_{N (G)}^{u}(a) = 0 ⇔ a = 0 for a ∈ N (G), a ≥ 0;

(iv) tr_{N (G)}^{u} is normal, i.e. for a monotone increasing net {a_{i} | i ∈ I} of positive elements a_{i} with supremum a we have tr_{N (G)}^{u}(a) = sup{tr (a_{i}) | i ∈ I}, or, equivalently, tr_{N (G)}^{u} is continuous with respect to the ultra-weak topology on N (G);

(v) || tr_{N (G)}^{u}(a)|| ≤ ||a|| for a ∈ N (G);

(vi) tr_{N (G)}^{u}(ab) = a tr_{N (G)}^{u}(b) for all a ∈ Z(N (G)) and b ∈ N (G);

(vii) Let p and q be projections in N (G). Then p ∼ q, i.e. p = uu^{∗} and q = u^{∗}u for some element u ∈ N (G), if and only if tr_{N (G)}^{u}(p) = tr_{N (G)}^{u}(q);

(viii) Any linear functional f: N (G) → ℂ which is continuous with respect to the norm topology on N (G) and which is central, i.e. f(ab) = f(ba) for all a, b ∈ N (G) factorizes as

N (G) ⟶^{tr_{N (G)}^{u}} Z(N (G)) ⟶^{f|_{Z(N (G))}} ℂ.

Definition 7.3 (Center valued dimension). For a finitely generated projective N (G)-module P define its center valued von Neumann dimension by

dim_{N (G)}^{u}(P) := ∑_{i=1}^{n} tr_{N (G)}^{u}(a_{i,i}) ∈ Z(N (G))^{ℤ/2} = {a ∈ Z(N (G)) | a = a^{∗}}

for any matrix A = (a_{i,j})_{i,j} ∈ M_{n}(N (G)) with A^{2} = A such that im(r_{A}: N (G)^{n} → N (G)^{n}) induced by right multiplication with A is N (G)-isomorphic to P.

There is a classification of von Neumann algebras into certain types. We only need to know what the type of a group von Neumann algebra is.

Lemma 7.4. Let G be a discrete group. Let G_{f} be the normal subgroup of G consisting of elements g ∈ G whose centralizer has finite index (or, equivalently, whose conjugacy class (g) consists of finitely many elements). Then:

(i) The group von Neumann algebra N (G) is of type I_{f} if and only if G is virtually abelian;

(ii) The group von Neumann algebra N (G) is of type II_{1} if and only if the index of G_{f} in G is infinite; (iii) Suppose that G is finitely generated. Then N (G) is of type I_{f} if G is

virtually abelian, and of type II_{1} if G is not virtually abelian;

(iv) The group von Neumann algebra N (G) is a factor, i.e. its center consists of {r · 1_{N (G)} | r ∈ ℂ}, if and only if G_{f} is the trivial group.

Proof. (i) This is proved in [61], [103]. (ii) This is proved in [61],[88]. (iii) This follows from assertions (i) and (ii) since for finitely generated G the group G_{f} has finite index in G if and only if G is virtually abelian. (iv) This follows from [23, Proposition 5 in III.7.6 on page 319].

The next result follows from [60, Theorem 8.4.3 on page 532, Theorem 8.4.4 on page 533].

Theorem 7.5 (K_{0} of finite von Neumann algebras). Let G be a group.

(i) The following statements are equivalent for two finitely generated projective N (G)-modules P and Q:

(a) P and Q are N (G)-isomorphic;

(b) P and Q are stably N (G)-isomorphic, i.e. P ⊕ V and Q ⊕ V are N (G)-isomorphic for some finitely generated projective N (G)-module V;

(c) dim_{N (G)}^{u}(P) = dim_{N (G)}^{u}(Q);

(d) [P] = [Q] in K_{0}(N (G));

(ii) The center valued dimension induces an injection

dim_{N (G)}^{u}: K_{0}(N (G)) → Z(N (G))^{ℤ/2} = {a ∈ Z(N (G)) | a = a^{∗}},

where the group structure on Z(N (G))^{ℤ/2} comes from addition. If N (G) is of type II_{1}, this map is an isomorphism.

Remark 7.6 (Group von Neumann algebras and representation theory). Theorem 7.5 shows that the group von Neumann algebra is the right generalization of the complex group ring from finite groups to infinite groups if one is concerned with representation theory of finite groups. Namely, let G be a finite group. Recall that a finite dimensional complex G-representation V is the same as a finitely generated ℂG-module and that K_{0}(ℂG) is the same as the complex representation ring. Moreover, two finite dimensional G-representations V and W are linearly G-isomorphic if and only if they have the same character. Recall that the character is a class function. One easily checks that the complex vector space of class functions on a finite group G is the same as the center Z(ℂG) and that the character of V contains the same information as dim_{N (G)}^{u}(V).

Remark 7.7 (Factors). Suppose that N (G) is a factor, i.e. its center consists of {r · 1_{N (G)} | r ∈ ℂ}. By Lemma 7.4 (iv) this is the case if and only if G_{f} is the trivial group. Then dim_{N (G)} = dim_{N (G)}^{u} and two finitely generated projective N (G)-modules P and Q are N (G)-isomorphic if and only if dim_{N (G)}(P) = dim_{N (G)}(Q) holds. This has the consequence that for a free G-CW -complex X of finite type the p-th L^{2}-Betti number determines the isomorphism type of PH_{p}^{G}(X; N (G)). In particular we must have PH_{p}^{G}(X; N (G)) ≅_{N (G)} N (G)^{n} provided that n = b_{p}^{(2)}(X; N (G)) is an integer. If one prefers to work with reduced L^{2}-homology, this is equivalent to the statement that H_{p}^{(2)}(X; l^{2}(G)) is isometrically G-linearly isomorphic to l^{2}(G)^{n} provided that n = b_{p}^{(2)}(X; N (G)) is an integer.

Remark 7.8 (The reduced L^{2}-cohomology of torsion-free groups). Let G be a torsion-free group. Suppose that it satisfies the Atiyah Conjecture 4.1 for (G, 1, ℚ). Suppose that there is a model for BG of finite type. Then we get for all p that PH_{p}^{G}(EG; N (G)) ≅_{N (G)} N (G)^{n}, or, equivalently, that H_{p}^{(2)}(X; l^{2}(G)) is isometrically G-linearly isomorphic to l^{2}(G)^{n} if the integer n is given by n = b_{p}^{(2)}(X; N (G)). This claim is proved in [80, solution to Exercise 10.11 on page 546].

We mention that the inclusion i: N (G) → U(G) induces an isomorphism

K_{0}(N (G)) ⟶^{≅} K_{0}(U(G)).

The Farrell-Jones Conjecture for K_{0}(ℂG), the Bass Conjecture and the passage in K_{0} from ℤG to ℂG and to N (G) is discussed in [80, Section 9.5.2] and [81].

7.2 The K_{1}-group and the L-groups of a Group von Neumann Algebra

A complete calculation of the K_{1}-group and of the L-groups of any von Neumann algebra and of the associated algebra of affiliated operators can be found in [80, Section 9.3 and Section 9.4], [83] and [99].

7.3 Applications to G-theory of Group Rings

Theorem 7.9 (Detecting G_{0}(ℂG) by K_{0}(N (G)) for amenable groups). If G is amenable, the map

l: G_{0}(ℂG) → K_{0}(N (G)), [M] ↦ [PN (G) ⊗_{ℂG} M]

is a well-defined homomorphism. If f: K_{0}(ℂG) → G_{0}(ℂG) is the forgetful map sending [P] to [P] and i_{∗}: K_{0}(ℂG) → K_{0}(N (G)) is induced by the inclusion i: ℂG → N (G), then the composition l ∘ f agrees with i_{∗}.

Proof. This is essentially a consequence of the dimension-flatness of N (G) over ℂG (see Theorem 5.1). Details of the proof can be found in [80, Theorem 9.64].

Now one can combine Theorem 7.5 and Theorem 7.9 to detect elements in G_{0}(ℂG) for amenable G. In particular one can show

dim_{ℚ} (ℚ ⊗_{ℤ} G_{0}(ℂG)) ≥ | con(G)_{f,cf}|,  (7.10)

where con(G)_{f,cf} is the set of conjugacy classes (g) of elements g ∈ G such that g has finite order and (g) contains only finitely many elements. Notice that con(G)_{f,cf} contains at least one element, namely the unit element e.

Remark 7.11 (The non-vanishing of [RG] in G_{0}(RG) for amenable groups). A direct consequence of Theorem 7.9 is that for an amenable group G the class [ℂG] in G_{0}(ℂG) generates an infinite cyclic subgroup. Namely, the dimension induces a well-defined homomorphism

dim_{N (G)} : G_{0}(ℂG) → ℝ, [M] ↦ dim_{N (G)} (N (G) ⊗_{ℂG} M) ,

which sends [ℂG] to 1. This result has been extended by Elek [32] to finitely generated amenable groups and arbitrary fields F, i.e. there is a well-defined homomorphism G_{0}(F G) → ℝ, which sends [F G] to 1 and is given by a certain rank function on finitely generated F G-modules.

The class [RG] in K_{0}(RG) is never zero for a commutative integral domain R with quotient field R_{(0)}. The augmentation RG → R and the map K_{0}(R) → ℤ, [P] ↦ dim_{R_{(0)}}(R_{(0)} ⊗_{R} P) together induce a homomorphism K_{0}(RG) → ℤ which sends [RG] to 1. A decisive difference between K_{0}(RG) and G_{0}(RG) is that [RG] = 0 is possible in G_{0}(RG) as the following example shows.

Example 7.12 (The vanishing of [RG] in G_{0}(RG) for groups G containing ℤ ∗ ℤ). We abbreviate F_{2} = ℤ ∗ ℤ. Suppose that G contains F_{2} as a subgroup. Let R be a ring. Then

[RG] = 0 ∈ G_{0}(RG)

holds by the following argument. Induction with the inclusion F_{2} → G induces a homomorphism G_{0}(RF_{2}) → G_{0}(RG) which sends [RF_{2}] to [RG]. Hence it suffices to show [RF_{2}] = 0 in G_{0}(RF_{2}). The cellular chain complex of the universal covering of S^{1} ∨ S^{1} yields an exact sequence of RF_{2}-modules 0 → (RF_{2})^{2} → RF_{2} → R → 0, where R is equipped with the trivial F_{2}-action. This implies [RF_{2}] = −[R] in G_{0}(RF_{2}). Hence it suffices to show [R] = 0 in G_{0}(RF_{2}). Choose an epimorphism f: F_{2} → ℤ. Restriction with f defines a homomorphism G_{0}(Rℤ) → G_{0}(RF_{2}). It sends the class of R viewed as trivial Rℤ-module to the class of R viewed as trivial RF_{2}-module. Hence it remains to show [R] = 0 in G_{0}(Rℤ). This follows from the exact sequence 0 → Rℤ ⟶^{s−1} Rℤ → R → 0 for s a generator of ℤ which comes from the cellular Rℤ-chain complex of S̃^{1}.

Remark 7.11 and Example 7.12 give some evidence for Conjecture 7.13. (Amenability and the regular representation in Gtheory). Let R be a commutative integral domain. Then a group G is amenable if and only if [RG] ≠ 0 in G_{0}(RG).

Remark 7.14 (The Atiyah Conjecture for amenable groups and G_{0}(ℂG)). Assume that G is amenable and that there is an upper bound on the orders of finite subgroups of G. Then the Atiyah Conjecture 4.1 for (G, d, ℂ) is true if and only if the image of the map

dim_{N (G)} : G_{0}(ℂG) → ℝ, [M] ↦ dim_{N (G)}(N (G) ⊗_{ℂG} M)

is contained in {r ∈ ℝ | d · r ∈ ℤ}.

Example 7.15 (K_{0}(ℂG) → G_{0}(ℂG) is not necessarily surjective). Let A = ⨁_{n∈ℤ} ℤ/2. This abelian group is locally finite. Hence the map

⨁_{H⊆A,|H|<∞} K_{0}(ℂH) → K_{0}(ℂA)

is surjective and the image of

dim_{N (G)} : K_{0}(ℂA) → ℝ, [P] ↦ dim_{N (A)} (N (A) ⊗_{ℂA} P)

is ℤ[1/2]. On the other hand the argument in [80, Example 10.13] shows that the map

dim_{N (G)} : G_{0}(ℂA) → ℝ, [M] ↦ dim_{N (A)} (N (A) ⊗_{ℂA} M)

is surjective. In particular the obvious map K_{0}(ℂA) → G_{0}(ℂA) is not surjective.

7.4 Applications to the Whitehead Group

The Whitehead group Wh(G) of a group G is the quotient of K_{1}(ℤG) by the subgroup which consists of elements given by units of the shape ±g ∈ ℤG for g ∈ G. Let i: H → G be the inclusion of a normal subgroup H ⊆ G. It induces a homomorphism i_{0} : Wh(H) → Wh(G). The conjugation action of G on H and on G induces a G-action on Wh(H) and on Wh(G) which turns out to be trivial on Wh(G). Hence i_{0} induces homomorphisms

i_{1} : ℤ ⊗_{ℤG} Wh(H) → Wh(G);  (7.16)
i_{2} : Wh(H)^{G} → Wh(G).  (7.17)

Theorem 7.18 (Detecting elements in Wh(G)). Let i: H → G be the inclusion of a normal finite subgroup H into an arbitrary group G. Then the maps i_{1} and i_{2} defined in (7.16) and (7.17) have finite kernel.

Proof. See [80, Theorem 9.38].

We emphasize that Theorem 7.18 above holds for all groups G. It seems to be related to the Farrell-Jones Isomorphism Conjecture.

8 L^{2}-Betti Numbers and Measurable Group Theory

In this section we want to discuss an interesting relation between L^{2}-Betti numbers and measurable group theory. We begin with formulating the main result.

Definition 8.1 (Measure equivalence). Two countable groups G and H are called measure equivalent if there exist commuting measure-preserving free actions of G and H on some standard Borel space (Ω, μ) with non-zero Borel measure μ such that the actions of both G and H admit measure fundamental domains X and Y of finite measure.

The triple (Ω, X, Y) is called a measure coupling of G and H. The index of (Ω, X, Y) is the quotient \frac{μ(X)}{μ(Y)}.

Here are some explanations. A Polish space is a separable topological space which is metrizable by a complete metric. A measurable space Ω = (Ω, A) is a set Ω together with a σ-algebra A. It is called a standard Borel space if it is isomorphic to a Polish space with its Borel σ-algebra. (The Polish space is not part of the structure, only its existence is required.) More information about this notion of measure equivalence can be found for instance in [36], [37] and [46, 0.5E].

The following result is due to Gaboriau [38, Theorem 6.3]. We will discuss its applications and sketch the proof based on homological algebra and the dimension function due to R. Sauer [100].

Theorem 8.2 (Measure equivalence and L^{2}-Betti numbers). Let G and H be two countable groups which are measure equivalent. If C > 0 is the index of a measure coupling, then we get for all p ≥ 0

b_{p}^{(2)}(G) = C · b_{p}^{(2)}(H).

The general strategy of the proof of Theorem 8.2 is as follows. In the first step one introduces the notion of a standard action G ↷ X and of a weak orbit equivalence of standard actions of index C and shows that two groups G and H are measure equivalent of index C if and only if there exist standard actions G ↷ X and H ↷ Y which are weakly orbit equivalent with index C. In the second step one assigns to a standard action G ↷ X L^{2}-Betti numbers b_{p}^{(2)}(G ↷ X), which involve only data that is invariant under orbit equivalence. Hence b_{p}^{(2)}(G ↷ X) itself depends only on the orbit equivalence class of G ↷ X. In order to deal with weak orbit equivalences, one has to investigate the behaviour of the L^{2}-Betti numbers of b_{p}^{(2)}(G ↷ X) under restriction. Finally one proves that the L^{2}-Betti numbers of a standard action G ↷ X agree with the L^{2}-Betti numbers of G itself.

A version of Theorem 8.2 for the L^{2}-torsion is presented in Conjecture 11.30.

8.1 Measure Equivalence and Quasi-Isometry

Remark 8.3 (Measure equivalence is the measure theoretic version of quasi-isometry). The notion of measure equivalence can be viewed as the measure theoretic analogue of the metric notion of quasi-isometric groups. Namely, two finitely generated groups G_{0} and G_{1} are quasi-isometric if and only if there exist commuting proper (continuous) actions of G_{0} and G_{1} on some locally compact space such that each action has a cocompact fundamental domain [46, 0.2 C′_{2} on page 6].

Example 8.4 (Infinite amenable groups). Every countable infinite amenable group is measure equivalent to ℤ (see [94]). Since obviously all the L^{2}-Betti numbers of ℤ vanish, Theorem 8.2 implies the result of Cheeger and Gromov that all the L^{2}-Betti numbers of an infinite amenable group vanish.

Remark 8.5 (L^{2}-Betti numbers and quasi-isometry). If the finitely generated groups G_{0} and G_{1} are quasi-isometric and there exist finite models for BG_{0} and BG_{1}, then b_{p}^{(2)}(G_{0}) = 0 ⇔ b_{p}^{(2)}(G_{1}) = 0 holds (see [46, page 224], [95]). But in general it is not true that there is a constant C > 0 such that b_{p}^{(2)}(G_{0}) = C · b_{p}^{(2)}(G_{1}) holds for all p ≥ 0 (cf. [39, page 7], [46, page 233], [109]).

Remark 8.6 (Measure equivalence versus quasi-isometry). If F_{g} denotes the free group on g generators, then define G_{n} := (F_{3} × F_{3}) ∗ F_{n} for n ≥ 2. The groups G_{m} and G_{n} are quasi-isometric for m, n ≥ 2 (see [21, page 105 in IV-B.46], [109, Theorem 1.5]) and have finite models for their classifying spaces. One easily checks using Theorem 6.1 that b_{1}^{(2)}(G_{n}) = n and b_{2}^{(2)}(G_{n}) = 4.

Theorem 8.2 due to Gaboriau implies that G_{m} and G_{n} are measure equivalent if and only if m = n holds. Hence there are finitely presented groups which are quasi-isometric but not measure equivalent.

The converse is also true. The groups ℤ^{n} and ℤ^{m} are infinite amenable and hence measure equivalent. But they are not quasi-isometric for different m and n since n is the growth rate of ℤ^{n} and the growth rate is a quasi-isometry invariant.

Notice that Theorem 8.2 implies that the sign of the Euler characteristic of a group G is an invariant under measure equivalence, which is not true for quasi-isometry by the example of the groups G_{n} above.

Let the two groups G and H act on the same metric space X properly and cocompactly by isometries. If X is second countable and proper, then G and H are measure equivalent. [100, Theorem 2.36]. If X is a geodesic and proper, then G and H are quasi-isometric.

Remark 8.7 (Kazhdan’s property (T)). Kazhdan’s property (T) is an invariant under measure equivalence [36, Theorem 8.2]. There exist quasiisometric finitely generated groups G_{0} and G_{1} such that G_{0} has Kazhdan’s property (T) and G_{1} not (see [39, page 7]). Hence G_{0} and G_{1} are quasi-isometric but not measure equivalent.

The rest of this section is devoted to an outline of the proof of Theorem 8.2 due to R. Sauer [100] which is simpler and more algebraic than the original one of Gaboriau [38] and may have the potential to apply also to L^{2}-torsion.

8.2 Discrete Measured Groupoids

A groupoid is a small category in which all morphisms are isomorphisms. We will identify a groupoid \underline{G} with its set of morphisms. Then the set of objects \underline{G}^{0} can be considered as a subset of \underline{G} via the identity morphisms. There are four canonical maps,

source map s: \underline{G} → \underline{G}^{0}, (f: x → y) ↦ x;
target map t: \underline{G} → \underline{G}^{0}, (f: x → y) ↦ y;
inverse map i: \underline{G} → \underline{G}, f ↦ f^{−1};
composition ∘: \underline{G}^{2} → \underline{G}, (f, g) ↦ f ∘ g,

where \underline{G}^{2} is \{(f, g) ∈ \underline{G} × \underline{G} | s(f) = t(g)\}. We will often abbreviate f ∘ g by fg.

A discrete measurable groupoid is a groupoid \underline{G} equipped with the structure of a standard Borel space such that the inverse map and the composition are measurable maps and s^{−1}(x) is countable for all objects x ∈ \underline{G}^{0}. Then \underline{G}^{0} ⊆ \underline{G} is a Borel subset, the source and the target maps are measurable and t^{−1}(x) is countable for all objects x ∈ \underline{G}^{0}.

Let μ be a probability measure on \underline{G}^{0}. Then for each measurable subset A ⊆ \underline{G} the function

\underline{G}^{0} → ℂ, x ↦ |s^{−1}(x) ∩ A|

is measurable and we obtain a σ-finite measure μ_{s} on \underline{G} by

μ_{s}(A) := ∫_{\underline{G}_{0}} |s^{−1}(x) ∩ A| dμ(x).

It is called the left counting measure of μ. The right counting measure μ_{t} is defined analogously replacing the source map s by the target map t. We call μ invariant if μ_{s} = μ_{t}, or, equivalently, if i_{∗}μ_{s} = μ_{s}. A discrete measurable groupoid \underline{G} together with an invariant measure μ on \underline{G}^{0} is called a discrete measured groupoid. Given a Borel subset A ⊆ \underline{G}^{0} with μ(A) > 0, there is the restricted discrete measured groupoid \underline{G}|_{A} = s^{−1}(A)∩t^{−1}(A), which is equipped with the normalized measure \frac{1}{μ(A)} · μ|_{A}.

An isomorphism of discrete measured groupoids f: \underline{G} → \underline{H} is an isomorphisms of groupoids which preserves the measures. Given measurable subsets A ⊆ \underline{G}^{0} and B ⊆ \underline{H}^{0} such that t(s^{−1}(A)) and t(s^{−1}(B)) have full measure in \underline{G}^{0} and \underline{H}^{0} respectively, we call an isomorphism of discrete measured groupoids f: \underline{G}_{A} → \underline{H}_{B} a weak isomorphism of discrete measured groupoids.

Example 8.8 (Orbit equivalence relation). Consider the countable group G with an action G ↷ X on a standard Borel space X with probability measure μ by μ-preserving isomorphisms. The orbit equivalence relation

R(G ↷ X) := {(x, gx) | x ∈ X, g ∈ G} ⊆ X × X

becomes a discrete measured groupoid by the obvious groupoid structure and measure.

An action G ↷ X of a countable group G is called standard if X is a standard Borel measure space with a probability measure μ, the action is by μ-preserving Borel isomorphisms and the action is essentially free, i.e. the stabilizer of almost every x ∈ X is trivial. Every countable group G admits a standard action, which is given by the shift action on ∏_{g∈G}[0, 1]. Notice that this G-action is not free but essentially free.

Two standard actions G ↷ X and H ↷ Y are weakly orbit equivalent if there are Borel subsets A ⊆ X and B ⊆ Y, which meet almost every orbit and have positive measure in X and Y respectively, and a Borel isomorphism f: A → B, which preserves the normalized measures on A and B and satisfies

f(G · x ∩ A) = H · f(x) ∩ B

for almost all x ∈ A. If A has full measure in X and B has full measure in Y, then the two standard actions are called orbit equivalent. The map f is called a weak orbit equivalence or orbit equivalence respectively. The index of a weak orbit equivalence of f is the quotient \frac{μ(A)}{μ(B)}. The next result is due to Furman [37, Theorem 3.3].

Theorem 8.9 (Measure equivalence and weak orbit equivalence). Two countable groups are measure equivalent with respect to a measure coupling of index C > 0 if and only if there exist standard actions of G and H which are weakly orbit equivalent with index C.

8.3 Groupoid Rings

Let \underline{G} be a discrete measured groupoid with invariant measure μ on \underline{G}^{0}. For a function ϕ: \underline{G} → ℂ and x ∈ \underline{G}^{0} put

S(ϕ)(x) := |\{g ∈ \underline{G} | ϕ(g) ≠ 0, s(g) = x\} ∈ \{0, 1, 2 …\} ⨿ \{∞\};
T (ϕ)(x) := |\{g ∈ \underline{G} | ϕ(g) ≠ 0, t(g) = x\} ∈ \{0, 1, 2 …\} ⨿ \{∞\}.

Let μ\underline{G} = μ_{s} = μ_{t} be the measure on \underline{G} induced by μ. Let L^{∞}(\underline{G}) = L^{∞}(\underline{G}; μ\underline{G}) be the ℂ-algebra of equivalence classes of essentially bounded measurable functions \underline{G} → ℂ. Define L^{∞}(\underline{G}^{0}) = L^{∞}(\underline{G}^{0}; μ) analogously. Define the groupoid ring of \underline{G} as the subset

ℂ\underline{G} := \{ϕ ∈ L^{∞}(\underline{G}) | S(ϕ) and T (ϕ) are essentially bounded on \underline{G}\}. (8.10)

The addition comes from the pointwise addition in L^{∞}(\underline{G}). Multiplication comes from the convolution product

(ϕ · ψ)(g) = ∑_{\begin{matrix}g_{1},g_{2}∈\underline{G} \\ g_{2}∘g_{1}=g \end{matrix}} ϕ(g_{1}) · ψ(g_{2}).

An involution of rings on ℂ\underline{G} is defined by (ϕ^{∗})(g) := \overline{ϕ(i(g))}. Define the augmentation homomorphism ϵ: ℂ\underline{G} → L^{∞}(\underline{G}^{0}) by sending ϕ to ϵ(ϕ): \underline{G}^{0} → ℂ, x ↦ ∑_{g∈s^{−1}(x)} ϕ(g). Notice that ϵ is in general not a ring homomorphism, it is only compatible with the additive structure. It becomes a homomorphism of ℂ\underline{G}-modules if we equip L^{∞}(\underline{G}^{0}) with the following ℂ\underline{G}-module structure

ϕ · f := ϵ(ϕ · j(f)) for ϕ ∈ ℂ\underline{G}, f ∈ L^{∞}(\underline{G}^{0}),

where j: L^{∞}(\underline{G}^{0}) → ℂ\underline{G} is the inclusion of rings, which is given by extending a function on \underline{G}^{0} to \underline{G} by putting it to be zero outside \underline{G}^{0}.

Given a group G and a ring R together with a homomorphism c: G → aut(R), define the crossed product ring R ∗_{c} G as the free R-module with G as R-basis and the multiplication given by

(∑_{g∈G} r_{g} · g) · (∑_{g∈G} s_{g} · g) = ∑_{g∈G} ( ∑_{\begin{matrix}g_{1},g_{2}∈G, \\ g=g_{1}g_{2} \end{matrix}} r_{g_{1}} · c(g_{1})(s_{g_{2}})) · g.

Given a standard action G ↷ X, let L^{∞}(X) ∗ G be the crossed product ring L^{∞}(X) ∗_{c} G with respect to the group homomorphism c: G → aut (L^{∞}(X)) sending g to the automorphism given by composition with l_{g^{−1}} : X → X, x ↦ g^{−1}x. We obtain an injective ring homomorphism

k: L^{∞}(X) ∗ G → ℂR(G ↷ X)

which sends ∑_{g∈G} f_{g} · g to the function (gx, x) ↦ f_{g}(gx). In the sequel we will regard L^{∞}(X) ∗ G as a subring of ℂR(G ↷ X) using k.

Next we briefly explain how one can associate to the groupoid ring ℂ\underline{G} of a discrete measured groupoid \underline{G} a von Neumann algebra N (\underline{G}), which is finite, or, equivalently, which possesses a faithful finite normal trace. One can define on ℂ\underline{G} an inner product

⟨ϕ, ψ⟩ = ∫\underline{G} ϕ(g) · \overline{ψ(g)} dμ\underline{G}.

Then ℂ\underline{G} as a ℂ-algebra with involution and the scalar product above satisfies the axioms of a Hilbert algebra A, i.e. we have ⟨y, x⟩ = ⟨x^{∗}, y^{∗}⟩ for x, y ∈ A, ⟨xy, z⟩ = ⟨y, x^{∗}z⟩ for x, y, z ∈ A and the map A → A, y ↦ yx is continuous for all x ∈ A. Let H_{A} be the Hilbert space completion of A with respect to the given inner product. Define the von Neumann algebra N (A) associated to A by the ℂ-algebra with involution B(H_{A})^{A} which consists of all bounded left A-invariant operators H_{A} → H_{A}. The standard trace is given by

tr_{N (A)} : N (A) → ℂ, f ↦ ⟨f(1_{A}), 1_{A}⟩ .

We do get a dimension function as in Theorem 1.11 for N (A).

Our main example will be N (G ↷ X) := N (ℂR(G ↷ X)) for a standard action G ↷ X of G.

If G is a countable group and \underline{G} = G is the associated discrete measured groupoid with one object, then ℂ\underline{G} = ℂG, l^{2}(G) = H_{ℂ\underline{G}} and the definition of N (\underline{G}) and tr_{N (\underline{G})} above agrees with the previous Definition 1.1 of N (G) and tr_{N (G)}.

Remark 8.11 (Summary and Relevance of the algebraic structures associated to a standard action). Let G ↷ X be a standard action. We have the following commutative diagram of inclusions of rings

ℂ ⟶  ℂG  ⟶^{=}  ℂG  ⟶ N (G)
↓  ↓  ↓  ↓
L^{∞}(X) ⟶ L^{∞}(X) ∗ G ⟶ ℂR(G ↷ X) ⟶ N (G ↷ X)

There is a ℂ\underline{G}-module structure on L^{∞}(R(G ↷ X)^{0}) = L^{∞}(X). Its restriction to L^{∞}(X) ∗ G ⊆ ℂR(G ↷ X) is the obvious L^{∞}(X) ∗ G-module structure on L^{∞}(X).

The following observation will be crucial. Given two standard actions G ↷ X and G ↷ Y, an orbit equivalence f from G ↷ X to H ↷ Y induces isomorphisms of rings, all denoted by f_{∗}, such that the following diagram with inclusions as horizontal maps commutes

L^{∞}(X) ⟶ ℂR(G ↷ X) ⟶ N (G ↷ X)
f_{∗}↓≅  ^{f_{∗}}↓≅  ^{f_{∗}}↓≅
L^{∞}(Y) ⟶ ℂR(H ↷ Y) ⟶ N (G ↷ Y)

It is not true that f induces a ring map L^{∞}(X)∗G → L^{∞}(Y)∗H, since we only require that f maps orbits to orbits but nothing is demanded about equivariance of f with respect to some homomorphism of groups from G → H. The crossed product ring L^{∞}(X) ∗ G contains too much information about the group G itself. Hence we shall only involve L^{∞}(X), ℂR(G ↷ X), and N (G ↷ X) in any algebraic construction which is designed to be invariant under orbit equivalence.

8.4 L^{2}-Betti Numbers of Standard Actions

Definition 8.12. Let \underline{G} be a discrete measured groupoid. Define its p-th L^{2}- Betti number by

b_{p}^{(2)}(\underline{G}) = dim_{N (\underline{G})} (Tor_{p}^{ℂ\underline{G}} (N (\underline{G}), L^{∞}(\underline{G}^{0}))) .

Given a standard action G ↷ X, define its p-th L^{2}-Betti number as the p-th L^{2}-Betti number of the associated orbit equivalence relation R(G ↷ X), i.e.

b_{p}^{(2)}(G ↷ X) = dim_{N (G↷X)} (Tor_{p}^{ℂR(G↷X)} (N (G ↷ X), L^{∞}(X))) .

Notice that Theorem 8.2 is true if we can prove the following three lemmas.

Lemma 8.13. If two standard actions G ↷ X and H ↷ Y are orbit equivalent, then they have the same L^{2}-Betti numbers.

Lemma 8.14. Let \underline{G} be a discrete measured groupoid. Let A ⊆ \underline{G} be a Borel subset such that t(s^{−1}(A)) has full measure in \underline{G}^{0}. Then we get for all p ≥ 0

b_{p}^{(2)}(\underline{G}) = μ(A) · b_{p}^{(2)}(\underline{G}|_{A}).

Lemma 8.15. Let G ↷ X be a standard action. Then we get for all p ≥ 0

b_{p}^{(2)}(G ↷ X) = b_{p}^{(2)}(G).

Lemma 8.13 follows directly from Remark 8.11. The hard part of the proof of Theorem 8.2 is indeed the proof of the remaining two Lemmas 8.14 and 8.15. This is essentially done by developing some homological algebra over finite von Neumann algebras taking the dimension for arbitrary modules into account.

8.5 Invariance of L^{2}-Betti Numbers under Orbit Equivalence

As an illustration we sketch the proof of Lemma 8.15. It follows from the following chain of equalities which we explain briefly below.

b_{p}^{(2)}(G) = dim_{N (G)} (Tor_{p}^{ℂG} (N (G), ℂ))  (8.16)
= dim_{N (G↷X)} (N (G ↷ X) ⊗_{N (G)} Tor_{p}^{ℂG} (N (G), ℂ))  (8.17)
= dim_{N (G↷X)} (Tor_{p}^{ℂG} (N (G ↷ X), ℂ))  (8.18)
= dim_{N (G↷X)} (Tor_{p}^{L^{∞}(X)∗G} (N (G ↷ X), L^{∞}(X) ∗ G ⊗_{ℂG} ℂ))(8.19)
= dim_{N (G↷X)} (Tor_{p}^{L^{∞}(X)∗G} (N (G ↷ X), L^{∞}(X)))  (8.20)
= dim_{N (G↷X)} (Tor_{p}^{ℂR(G↷X)} (N (G ↷ X), L^{∞}(X)))  (8.21)
= b_{p}^{(2)}(G ↷ X).  (8.22)

Equations (8.16) and (8.22) are true by definition. The inclusion of von Neumann algebras N (G) → N (G ↷ X) preserves the traces. This implies that the functor N (G ↷ X) ⊗_{N (G)} − from N (G)-modules to N (G ↷ X)-modules is faithfully flat and preserves dimensions. The proof of this fact is completely analogous to the proof of Theorem 1.18. This shows (8.17) and (8.18). For every ℂG-module M there is a natural L^{∞}(X) ∗ G-isomorphism

L^{∞}(X) ∗ G ⊗_{ℂG} M ⟶^{≅} L^{∞}(X) ⊗_{ℂ} M.

This shows that L^{∞}(X) ∗ G is flat as ℂG-module and that (8.19) and (8.20) are true. The hard part is now to prove (8.21), which is the decisive step, since here one eliminates L^{∞}(X) ∗ G from the picture and stays with terms which depend only on the orbit equivalence class of G ↷ X. Its proof involves homological algebra and dimension theory. It is not true that the relevant Tor-terms are isomorphic, they only have the same dimension.

This finishes the outline of the proof of Lemma 8.15 and of Theorem 8.2. The complete proof can be found in [100].

9 The Singer Conjecture

In this section we briefly discuss the following conjecture.

Conjecture 9.1 (Singer Conjecture). If M is an aspherical closed manifold, then

b_{p}^{(2)}(M̃) = 0 if 2p ≠ dim(M).

If M is a closed connected Riemannian manifold with negative sectional curvature, then

b_{p}^{(2)}(M̃) \{ \begin{matrix}= 0 & if 2p ≠ dim(M); \\ > 0 & if 2p = dim(M). \end{matrix}

We mention that all the explicit computations presented in Section 3 are compatible with the Singer Conjecture 9.1. A version of the Singer Conjecture for L^{2}-torsion will be presented in Conjecture 11.28.

9.1 The Singer Conjecture and the Hopf Conjecture

Because of the Euler-Poincaré formula χ(M) = ∑_{p≥0}(−1)^{p} · b_{p}^{(2)}(M̃) (see Theorem 2.7 (x)) the Singer Conjecture 9.1 implies the following conjecture in case M is aspherical or has negative sectional curvature.

Conjecture 9.2 (Hopf Conjecture). If M is an aspherical closed manifold of even dimension, then

(−1)^{dim(M)/2} · χ(M) ≥ 0.

If M is a closed Riemannian manifold of even dimension with sectional curvature sec(M), then

(−1)^{dim(M)/2} · χ(M) > 0  if sec(M) < 0;
(−1)^{dim(M)/2} · χ(M) ≥ 0  if sec(M) ≤ 0;
χ(M) = 0  if sec(M) = 0;
χ(M) ≥ 0  if sec(M) ≥ 0;
χ(M) > 0  if sec(M) > 0.

In original versions of the Singer Conjecture 9.1 and the Hopf Conjecture 9.2 the statements for aspherical manifolds did not appear. Every Riemannian manifold with non-positive sectional curvature is aspherical by Hadamard’s Theorem.

9.2 Pinching Conditions

The following two results are taken from the paper by Jost and Xin [59, Theorem 2.1 and Theorem 2.3].

Theorem 9.3. Let M be a closed connected Riemannian manifold of dimension dim(M) ≥ 3. Suppose that there are real numbers a > 0 and b > 0 such that the sectional curvature satisfies −a^{2} ≤ sec(M) ≤ 0 and the Ricci curvature is bounded from above by −b^{2}. If the non-negative integer p satisfies 2p ≠ dim(M) and 2pa ≤ b, then

b_{p}^{(2)}(M̃) = 0.

Theorem 9.4. Let M be a closed connected Riemannian manifold of dimension dim(M) ≥ 4. Suppose that there are real numbers a > 0 and b > 0 such that the sectional curvature satisfies −a^{2} ≤ sec(M) ≤ −b^{2}. If the non-negative integer p satisfies 2p ≠ dim(M) and (2p − 1) · a ≤ (dim(M) − 2) · b, then

b_{p}^{(2)}(M̃) = 0.

The next result is a consequence of a result of Ballmann and Brüning [3, Theorem B on page 594].

Theorem 9.5. Let M be a closed connected Riemannian manifold. Suppose that there are real numbers a > 0 and b > 0 such that the sectional curvature satisfies −a^{2} ≤ sec(M) ≤ −b^{2}. If the non-negative integer p satisfies 2p < dim(M) − 1 and p · a < (dim(M) − 1 − p) · b, then

b_{p}^{(2)}(M̃) = 0.

Theorem 9.4 and Theorem 9.5 are improvements of the older results by Donnelly and Xavier [26].

Remark 9.6 (Right angled Coxeter groups and Coxeter complexes). Next we mention the work of Davis and Okun [20]. A simplicial complex L is called a flag complex if each finite non-empty set of vertices which pairwise are connected by edges spans a simplex of L. To such a flag complex they associate a right-angled Coxeter group W_{L} defined by the following presentation [20, Definition 5.1]. Generators are the vertices v of L. Each generator v satisfies v^{2} = 1. If two vertices v and w span an edge, there is the relation (vw)^{2} = 1. Given a finite flag complex L, Davis and Okun associate to it a finite proper W_{L}- CW -complex Σ_{L}, which turns out to be a model for the classifying space of the family of finite subgroups E_{FIN} (W_{L}) [20, 6.1, 6.1.1 and 6.1.2]. Equipped with a specific metric, Σ_{L} turns out to be non-positive curved in a combinatorial sense, namely, it is a CAT(0)-space [20, 6.5.3]. If L is a generalized rational homology (n−1)-sphere, i.e. a homology (n−1)-manifold with the same rational homology as S^{n−1}, then Σ_{L} is a polyhedral homology n-manifold with rational coefficients [20, 7.4]. So Σ_{L} is a reminiscence of the universal covering of a closed n-dimensional manifold with non-positive sectional curvature and fundamental group W_{L}. In view of the Singer Conjecture 9.1 the conjecture makes sense that b_{p}^{(2)}(Σ_{L}; N (W_{L})) = 0 for 2p ≠ n provided that the underlying topological space of L is S^{n−1} (or, more generally, that it is a homology (n − 1)-sphere) [20, Conjecture 0.4 and 8.1]. Davis and Okun show that the conjecture is true in dimension n ≤ 4 and that it is true in dimension (n + 1) if it holds in dimension n and n is odd [20, Theorem 9.3.1 and Theorem 10.4.1].

9.3 The Singer Conjecture and Kähler Manifolds

Definition 9.7. Let (M, g) be a connected Riemannian manifold. A (p−1)-form η ∈ Ω^{p−1}(M) is bounded if ||η||_{∞} := sup{||η||_{x} | x ∈ M} < ∞ holds, where ||η||_{x} is the norm on Alt^{p−1}(T_{x}M) induced by g_{x}. A p-form ω ∈ Ω^{p}(M) is called d(bounded) if ω = d(η) holds for some bounded (p − 1)-form η ∈ Ω^{p−1}(M). A p-form ω ∈ Ω^{p}(M) is called d̃(bounded) if its lift ω̃ ∈ Ω^{p}(M̃) to the universal covering M̃ is d(bounded).

The next definition is taken from [45, 0.3 on page 265].

Definition 9.8 (Kähler hyperbolic manifold). A Kähler hyperbolic manifold is a closed connected Kähler manifold (M, h) whose fundamental form ω is d̃(bounded).

Example 9.9 (Examples of Kähler hyperbolic manifolds). The following list of examples of Kähler hyperbolic manifolds is taken from [45, Example 0.3]:

(i) M is a closed Kähler manifold which is homotopy equivalent to a Riemannian manifold with negative sectional curvature;

(ii) M is a closed Kähler manifold such that π_{1}(M) is word-hyperbolic in the sense of [44] and π_{2}(M) = 0;

(iii) M̃ is a symmetric Hermitian space of non-compact type;

(iv) M is a complex submanifold of a Kähler hyperbolic manifold;

(v) M is a product of two Kähler hyperbolic manifolds.

The following result is due to Gromov [44, Theorem 1.2.B and Theorem 1.4.A on page 274]. A detailed discussion of the proof and the consequences of this theorem can also be found in [80, Chapter 11].

Theorem 9.10. (L^{2}-Betti numbers of Kähler hyperbolic manifolds). Let M be a Kähler hyperbolic manifold of complex dimension m and real dimension n = 2m. Then

b_{p}^{(2)}(M̃) = 0  if p ≠ m;
b_{m}^{(2)}(M̃) > 0;
(−1)^{m} · χ(M) > 0.

10 The Approximation Conjecture

This section is devoted to the following conjecture.

Conjecture 10.1 (Approximation Conjecture). A group G satisfies the Approximation Conjecture if the following holds:

Let {G_{i} | i ∈ I} be an inverse system of normal subgroups of G directed by inclusion over the directed set I. Suppose that ⋂_{i∈I} G_{i} = {1}. Let X be a G-CW -complex of finite type. Then G_{i}\X is a G/G_{i}-CW -complex of finite type and

b_{p}^{(2)}(X; N (G)) = lim_{i∈I} b_{p}^{(2)}(G_{i}\X; N (G/G_{i})).

Remark 10.2 (The Approximation Conjecture for subgroups of finite index). Let us consider the special case where the inverse system {G_{i} | i ∈ I} is given by a nested sequence of normal subgroups of finite index

G = G_{0} ⊃ G_{1} ⊃ G_{2} ⊃ G_{3} ⊃ ….

Notice that then b_{p}^{(2)}(G_{i}\X; N (G/G_{i})) = \frac{b_{p}(G_{i}\X)}{[G:G_{i}]}, where b_{p}(G_{i}\X) is the classical p-th Betti number of the finite CW -complex G_{i}\X. In this special case Conjecture 10.1 was formulated by Gromov [46, pages 20, 231] and proved in [72, Theorem 0.1]. Thus we get an asymptotic relation between the L^{2}-Betti numbers and Betti numbers, namely

b_{p}^{(2)}(X; N (G)) = lim_{i→∞} \frac{b_{p}(G_{i}\X)}{[G : G_{i}]},

although the Betti numbers of a connected finite CW -complex Y and the L^{2}- Betti numbers of its universal covering Ỹ have nothing in common except the fact that their alternating sum equals χ(Y) (see Example 6.2).

Interesting variations of this result for not necessarily normal subgroups of finite index and Betti-numbers with coefficients in representations can be found in the paper by Farber [34].

Definition 10.3. Let G be the smallest class of groups which contains the trivial group and is closed under the following operations:

(i) Amenable quotient Let H ⊆ G be a (not necessarily normal) subgroup. Suppose that H ∈ G and the quotient G/H is an amenable discrete homogeneous space. (For the precise definition of amenable discrete homogeneous space see for instance [80, Definition 13.8]. If H ⊆ G is normal and G/H is amenable, then G/H is an amenable discrete homogeneous space.)

Then G ∈ G;

(ii) Colimits If G = colim_{i∈I} G_{i} is the colimit of the directed system {G_{i} | i ∈ I} of groups indexed by the directed set I and each G_{i} belongs to G, then G belongs to G;

(iii) Inverse limits If G = lim_{i∈I} G_{i} is the limit of the inverse system {G_{i} | i ∈ I} of groups indexed by the directed set I and each G_{i} belongs to G, then G belongs to G;

(iv) Subgroups If H is isomorphic to a subgroup of the group G with G ∈ G, then H ∈ G;

(v) Quotients with finite kernel Let 1 → K → G → Q → 1 be an exact sequence of groups. If K is finite and G belongs to G, then Q belongs to G.

Next we provide some information about the class G. Notice that in the original definition of G due to Schick [102, Definition 1.12] the resulting class is slightly smaller: there it is required that the class contains the trivial subgroup and is closed under operations (i), (ii), (iii) and (iv), but not necessarily under operation (v). The proof of the next lemma can be found in [80, Lemma 13.11].

Lemma 10.4. (i) A group G belongs to G if and only if every finitely generated subgroup of G belongs to G;

(ii) The class G is residually closed, i.e. if there is a nested sequence of normal subgroups G = G_{0} ⊃ G_{1} ⊃ G_{2} ⊃ … such that ⋂_{i≥0} G_{i} = {1} and each quotient G/G_{i} belongs to G, then G belongs to G;

(iii) Any residually amenable and in particular any residually finite group belongs to G;

(iv) Suppose that G belongs to G and f: G → G is an endomorphism. Define the “mapping torus group” G_{f} to be the quotient of G ∗ ℤ obtained by introducing the relations t^{−1}gt = f(g) for g ∈ G and t ∈ ℤ a fixed generator. Then G_{f} belongs to G;

(v) Let {G_{j} | j ∈ J} be a set of groups with G_{j} ∈ G. Then the direct sum ⨁_{j∈J} G_{j} and the direct product ∏_{j∈J} G_{j} belong to G.

The proof of the next result can be found in [80, Theorem 13.3]. It is a mild generalization of the results of Schick [101] and [102], where the original proof of the Approximation Conjecture for subgroups of finite index was generalized to the much more general setting above and then applied to the Atiyah Conjecture. The connection between the Approximation Conjecture and the Atiyah Conjecture for torsion-free groups comes from the obvious fact that a convergent series of integers has an integer as limit.

Theorem 10.5 (Status of the Approximation Conjecture). Every group G which belongs to the class G (see Definition 10.3) satisfies the Approximation Conjecture 10.1.

11 L^{2}-Torsion

Recall that L^{2}-Betti numbers are modelled on Betti numbers. Analogously one can generalize the classical notion of Reidemeister torsion to an L^{2}-setting, which will lead to the notion of L^{2}-torsion. The L^{2}-torsion may be viewed as a secondary L^{2}-Betti number just as the Reidemeister torsion can be viewed as a secondary Betti number. Namely, the Reidemeister torsion is only defined if all the Betti numbers (with coefficients in a suitable representation) vanish, and similarly the L^{2}-torsion is defined only if the L^{2}-Betti numbers vanish. Both invariants give valuable information about the spaces in question.

11.1 The Fuglede-Kadison Determinant

In this subsection we briefly explain the notion of the Fuglede-Kadison determinant. We have extended the notion of the (classical) dimension of a finite dimensional complex vector space to the von Neumann dimension of a finitely generated projective N (G)-module (and later even to arbitrary N (G)-modules). Similarly we want to generalize the classical determinant of an endomorphism of a finite dimensional complex vector space to the Fuglede-Kadison determinant of an N (G)-endomorphism f: P → P of a finitely generated projective N (G)- module P and of an N (G)-map f: N (G)^{m} → N (G)^{n} of based finitely generated N (G)-modules. This is necessary since for the definition of Reidemeister torsion one needs determinants and hence for the definition of L^{2}-torsion one has to develop an appropriate L^{2}-analogue.

Definition 11.1 (Spectral density function). Let f: N (G)^{m} → N (G)^{n} be an N (G)-homomorphism. Let ν(f): l^{2}(G)^{m} → l^{2}(G)^{n} be the associated bounded G-equivariant operator (see Remark 1.7). Denote by {E_{λ}^{f^{∗}f} | λ ∈ ℝ} the (rightcontinuous) family of spectral projections of the positive operator ν(f^{∗}f). Define the spectral density function of f by

F_{f}: ℝ → [0, ∞) λ ↦ dim_{N (G)} (im(E_{λ^{2}}^{f^{∗}f})) .

The spectral density function is monotone and right-continuous. It takes values in [0, m]. Here and in the sequel |x| denotes the norm of an element x of a Hilbert space and ∥T ∥ the operator norm of a bounded operator T. Since ν(f) and ν(f^{∗}f) have the same kernel, dim_{N (G)}(ker(f)) = F_{f}(0).

Example 11.2 (Spectral density function for finite G). Suppose that G is finite. Then ℂG = N (G) = l^{2}(G) and ν(f) = f. Let 0 ≤ λ_{0} < … < λ_{r} be the eigenvalues of f^{∗}f and μ_{i} be the multiplicity of λ_{i}, i.e. the dimension of the eigenspace of λ_{i}. Then the spectral density function is a right continuous step function which is zero for λ < 0 and has a step of height \frac{μ_{i}}{|G|} at each \sqrt{λ_{i}}.

Example 11.3 (Spectral density function for G = ℤ^{n}). Let G = ℤ^{n}. We use the identification N (ℤ^{n}) = L^{∞}(T^{n}) of Example 1.4. For f ∈ L^{∞}(T^{n}) the spectral density function F_{M_{f}} of M_{f}: L^{2}(T^{n}) → L^{2}(T^{n}), g ↦ g · f sends λ to the volume of the set {z ∈ T^{n} | |f(z)| ≤ λ}.

Definition 11.4. (Fuglede-Kadison determinant of N (G)-maps N (G)^{m} → N (G)^{n}). Let f: N (G)^{m} → N (G)^{n} be an N (G)-map. Let F_{f}(λ) be the spectral density function of Definition 11.1 which is a monotone non-decreasing rightcontinuous function. Let dF be the unique measure on the Borel σ-algebra on ℝ which satisfies dF (]a, b]) = F (b) − F (a) for a < b. Then define the Fuglede-Kadison determinant

det_{N (G)}(f) ∈ [0, ∞)

by the positive real number

det_{N (G)}(f) = exp (∫_{0+}^{∞} ln(λ) dF )

if the Lebesgue integral ∫_{0+}^{∞} ln(λ) dF converges to a real number and by 0 otherwise.

Notice that in the definition above we do not require m = n or that f is injective or f is surjective.

Example 11.5 (Fuglede-Kadison determinant for finite G). To illustrate this definition, we look at the example where G is finite. We essentially get the classical determinant det_{ℂ}. Namely, we have computed the spectral density function for finite G in Example 11.2. Let λ_{1}, λ_{2}, …, λ_{r} be the non-zero eigenvalues of f^{∗}f with multiplicity μ_{i}. Then one obtains, if \overline{f^{∗}f} is the automorphism of the orthogonal complement of the kernel of f^{∗}f induced by f^{∗}f,

det_{N (G)}(f) = exp (∑_{i=1}^{r} \frac{μ_{i}}{|G|} · ln(\sqrt{λ_{i}})) = ∏_{i=1}^{r} λi^{\frac{μ_{i}}{2·|G|}} = (det_{ℂ} (\overline{f^{∗}f}))^{\frac{1}{2·|G|}} .

If f: ℂG^{m} → ℂG^{m} is an automorphism, we get

det_{N (G)}(f) = |det_{ℂ}(f)|^{\frac{1}{|G|}} .

Example 11.6 (Fuglede-Kadison determinant over N (ℤ^{n})). Let G = ℤ^{n}. We use the identification N (ℤ^{n}) = L^{∞}(T^{n}) of Example 1.4. For f ∈ L^{∞}(T^{n}) we conclude from Example 11.3

det_{N (ℤ^{n})} (M_{f}: L^{2}(T^{n}) → L^{2}(T^{n})) = exp (∫_{T^{n}} ln(|f(z)|) · χ_{{u∈S^{1}|f(u)≠0}} dvol_{z})

using the convention exp(−∞) = 0.

Here are some basic properties of this notion. A morphism f: N (G)^{m} → N (G)^{n} has dense image if the closure \overline{im(f)} of its image in N (G)^{n} in the sense of Definition 1.10 is N (G)^{n}. The adjoint A^{∗} of a matrix A = (a_{i,j}) ∈ M(m, n; N (G)) is the matrix in M(n, m; N (G)) given by (a_{j,i}^{∗}), where ∗: N (G) → N (G) sends an operator a to its adjoint a^{∗}. The adjoint f^{∗}: N (G)^{n} → N (G)^{m} of f: N (G)^{m} → N (G)^{n} is given by the matrix A^{∗} if f is given by the matrix A. The proof of the next result can be found in [80, Theorem 3.14].

Theorem 11.7 (Fuglede-Kadison determinant).

(i) Composition Let f: N (G)^{l} → N (G)^{m} and g: N (G)^{m} → N (G)^{n} be N (G)-homomor-phisms such that f has dense image and g is injective. Then

det_{N (G)}(g ∘ f) = det_{N (G)}(g) · det_{N (G)}(f);

(ii) Additivity Let f_{1} : N (G)^{m_{1}} → N (G)^{n_{1}}, f_{2} : N (G)^{m_{2}} → N (G)^{n_{2}} and f_{3} : N (G)^{m_{3}} → N (G)^{n_{3}} be N (G)-homomorphisms such that f_{1} has dense image and f_{2} is injective. Then

det_{N (G)} ( \begin{matrix}f_{1} & f_{3} \\ 0 & f_{2} \end{matrix} ) = det_{N (G)}(f_{1}) · det_{N (G)}(f_{2});

(iii) Invariance under adjoint map

Let f: N (G)^{m} → N (G)^{n} be an N (G)-homomorphism. Then

det_{N (G)}(f) = det_{N (G)}(f^{∗});

(iv) Induction Let i: H → G be an injective group homomorphism and let f: N (H)^{m} → N (H)^{n} be anN (H)-homomorphism. Then

det_{N (G)}(i_{∗}f) = det_{N (H)}(f).

Definition 11.8. (Fuglede-Kadison determinant of N (G)-endomorphisms of finitely generated projective modules). Let f: P → P be an endomorphism of a finitely generated projective N (G)-module P. Choose a finitely generated projective N (G)-module Q and an N (G)-isomorphism u: N (G)^{n} ⟶^{≅} P ⊕Q. Define the Fuglede-Kadison determinant

det_{N (G)}(f) ∈ [0, ∞)

by the Fuglede-Kadison determinant in the sense of Definition 11.4

det_{N (G)} (u^{−1} ∘ (f ⊕ id_{Q}) ∘ u) .

This definition is independent of the choices of Q and u by Theorem 11.7. Notice that in Definition 11.8 no N (G)-basis appear but that it works only for endomorphisms, whereas in Definition 11.4 we work with finitely generated free based modules but do not require that the source and target of f are isomorphic. There is an obvious analogue of Theorem 11.7 for the Fuglede-Kadison determinant of endomorphisms of finitely generated projective N (G)- modules.

11.2 The Determinant Conjecture

It will be important for applications to geometry to study the Fuglede-Kadison determinant of N (G)-maps f: N (G)^{m} → N (G)^{n} which come by induction from ℤG-maps or ℂG-maps. The following example is taken from [80, Example 3.22].

Example 11.9 (Fuglede-Kadison determinant of maps coming from elements in ℂ[ℤ]). Consider a non-trivial element p ∈ ℂ[ℤ] = ℂ[z, z^{−1}]. We can write

p(z) = C · z^{n} · ∏_{k=1}^{l}(z − a_{k})

for non-zero complex numbers C, a_{1}, …, a_{l} and non-negative integers n, l. Let r_{p}: N (ℤ) → N (ℤ) be the N (ℤ)-map given by right multiplication with p. Then

det_{N (ℤ)}(r_{p}) = |C| · ∏_{\begin{matrix}1≤k≤l, \\ |a_{k}|>1 \end{matrix}} |a_{k}|.

Definition 11.10 (Determinant class). A group G is of det ≥ 1-class if for each A ∈ M(m, n; ℤG) the Fuglede-Kadison determinant (see Definition 11.4) of the morphism r_{A}: N (G)^{m} → N (G)^{n} given by right multiplication with A satisfies

det_{N (G)}(r_{A}) ≥ 1.

Conjecture 11.11 (Determinant Conjecture). Every group G is of det ≥ 1-class.

The proof of the next result can be found in [80, Theorem 13.3]. It is a mild generalization of the results of Schick [101] and [102].

Theorem 11.12 (Status of the Determinant Conjecture). Every group G which belongs to the class G (see Definition 10.3) satisfies the Determinant Conjecture 11.11.

One easily checks that the Fuglede-Kadison determinant defines a homomorphism of abelian groups

Φ^{G} : Wh(G) → (0, ∞) = {r ∈ ℝ | r > 0}  (11.13)

with respect to the group structure given by multiplication of positive real numbers on the target. We mention the following conjecture.

Conjecture 11.14 (Triviality of the map induced by the Fuglede-Kadison determinant on Wh(G)). The map Φ^{G}: Wh(G) → (0, ∞) is trivial.

Lemma 11.15. (i) If G satisfies the Determinant Conjecture 11.11, then G satisfies Conjecture 11.14;

(ii) The Approximation Conjecture 10.1 for G and the inverse system {G_{i} | i ∈ I} is true if each group G_{i} is of det ≥ 1-class.

Proof. See [80, Theorem 13.3 (1) and Lemma 13.6].

11.3 Definition and Basic Properties of L^{2}-Torsion

We will consider L^{2}-torsion only for universal coverings and in the L^{2}-acyclic case. A more general setting is treated in [80, Section 3.4].

Definition 11.16 (det-L^{2}-acyclic). Let X be a finite connected CW -complex with fundamental group π = π_{1}(X). Let C_{∗}^{N} (X̃) be the N (π)-chain complex N (G) ⊗_{ℤG} C_{∗}(X̃) with p-th differential c_{p}^{N} = id_{N (G)} ⊗_{ℤ}c_{p}. We say that X is det-L^{2}-acyclic if for each p we get for the Fuglede-Kadison determinant of c_{p}^{N} and for the p-th L^{2}-Betti number of X̃

det_{N (π)} (c_{p}^{N} ) > 0;
b_{p}^{(2)}(X̃) = 0.

If X is det-L^{2}-acyclic, we define the L^{2}-torsion of X̃ by

ρ^{(2)}(X̃) = − ∑_{p≥0}(−1)^{p} · ln (det_{N (G)} (c_{p}^{N} )) ∈ ℝ.

If X is a finite CW -complex, we call it det-L^{2}-acyclic if each component C is det-L^{2}-acyclic. In this case we define

ρ^{(2)}(X̃) := ∑_{C∈π_{0}(X)} ρ^{(2)}(C̃).

The condition that X̃ is L^{2}-acyclic is not needed for the definition of L^{2}- torsion, but is necessary to ensure the basic and useful properties which we will discuss below.

Remark 11.17 (L^{2}-torsion in terms of the Laplacian). One can express the L^{2}-torsion also in terms of the Laplacian which is closer to the notions of analytic torsion and analytic L^{2}-torsion. After a choice of cellular ℤπbasis, every N (π)-chain module C_{p}^{N} (X̃) looks like N (π)^{n_{p}} for appropriate nonnegative integers n_{p}. Hence we can assign to c_{p}^{N} : C_{p}^{N} (X̃) → C_{p−1}^{N}(X̃) its adjoint (c_{p}^{N} )^{∗} : C_{p−1}^{N}(X̃) → C_{p}^{N} (X̃) which is given by the matrix (a_{j,i}^{∗}) if c_{p}^{(2)} is given by the matrix (a_{i,j}). Define the p-th Laplace homomorphism Δ_{p}: C_{p}^{N} (X̃) → C_{p}^{N} (X̃) to be the N (π)-homomorphism (c_{p}^{N} )^{∗} ∘ c_{p}^{N} + c_{p+1}^{N} ∘ (c_{p+1}^{N})^{∗}. Then X̃ is det-L^{2}-acyclic if and only if Δ_{p} is injective and has dense image, i.e. the closure of its image in C_{p}^{N} (X̃) is C_{p}^{N} (X̃), and det_{N (π)}(Δ_{p}) > 0. In this case we get

ρ^{(2)}(X̃) = − \frac{1}{2} · ∑_{p≥0}(−1)^{p} · p · ln (det_{N (π)}(Δ_{p})) .

This follows from [80, Lemma 3.30].

The next theorem presents the basic properties of ρ^{(2)}(X̃) and is proved in [80, Theorem 3.96]. Notice the formal analogy between the behaviour of ρ^{(2)}(X̃) and the classical Euler characteristic χ(X).

Theorem 11.18. (Cellular L^{2}-torsion for universal coverings).

(i) Homotopy invariance Let f: X → Y be a homotopy equivalence of finite CW -complexes. Let τ(f) ∈ Wh(π_{1}(Y)) be its Whitehead torsion (see [17]). Suppose that X̃ or Ỹ is det-L^{2}-acyclic. Then both X̃ and Ỹ are det-L^{2}-acyclic and

ρ^{(2)}(Ỹ) − ρ^{(2)}(X̃) = Φ^{π_{1}(Y)}(τ(f)),

where Φ^{π_{1}(Y)} : Wh(π_{1}(Y)) = ⨁_{C∈π_{0}(Y)} Wh(π_{1}(C)) → ℝ is the sum of the maps Φ^{π_{1}(C)} of (11.13);

(ii) Sum formula Consider the pushout of finite CW -complexes such that j_{1} is an inclusion of CW -complexes, j_{2} is cellular and X inherits its CW -complex structure from X_{0}, X_{1} and X_{2}

X_{0} ⟶^{j_{1}} X_{1}
j_{2}↓  ↓^{i_{1}}
X_{2} ⟶_{i_{2}} X

Assume X̃_{0}, X̃_{1} and X̃_{2} are det-L^{2}-acyclic and that for k = 0, 1, 2 the map π_{1}(i_{k}): π_{1}(X_{k}) → π_{1}(X) induced by the obvious map i_{k}: X_{k} → X is injective for all base points in X_{k}.

Then X̃ is det-L^{2}-acyclic and we get

ρ^{(2)}(X̃) = ρ^{(2)}(X̃_{1}) + ρ^{(2)}(X̃_{2}) − ρ^{(2)}(X̃_{0});

(iii) Poincaré duality Let M be a closed manifold of even dimension. Equip it with some CW - complex structure. Suppose that M̃ is det-L^{2}-acyclic. Then

ρ^{(2)}(M̃) = 0;

(iv) Product formula

Let X and Y be finite CW -complexes. Suppose that X̃ is det-L^{2}-acyclic. Then { \atop X × Y} is det-L^{2}-acyclic and

ρ^{(2)}({ \atop X × Y}) = χ(Y) · ρ^{(2)}(X̃);

(v) Multiplicativity Let X → Y be a finite covering of finite CW -complexes with d sheets. Then X̃ is det-L^{2}-acyclic if and only if Ỹ is det-L^{2}-acyclic and in this case

ρ^{(2)}(X̃) = d · ρ^{(2)}(Ỹ).

The next three results are taken from [80, Corollary 3.103, Theorem 3.105 and Theorem 3.111]. There is also a more general version of Theorem 11.19 for fibrations (see [80, Theorem 3.100]).

Theorem 11.19 (L^{2}-torsion and fiber bundles). Suppose that F → E ⟶^{p} B is a (locally trivial) fiber bundle of finite CW -complexes with B connected. Suppose that for one (and hence all) b ∈ B the inclusion of the fiber F_{b} into E induces an injection on the fundamental groups for all base points in F_{b} and F̃_{b} is det-L^{2}-acyclic. Then Ẽ is det-L^{2}-acyclic and

ρ^{(2)}(Ẽ) = χ(B) · ρ^{(2)}(F̃).

Theorem 11.20 (L^{2}-torsion and S^{1}-actions). Let X be a connected S^{1}- CW -complex of finite type. Suppose that for one orbit S^{1}/H (and hence for all orbits) the inclusion into X induces a map on π_{1} with infinite image. (In particular the S^{1}-action has no fixed points.) Then X̃ is det-L^{2}-acyclic and

ρ^{(2)}(X̃) = 0.

Theorem 11.21 (L^{2}-torsion on aspherical closed S^{1}-manifolds). Let M be an aspherical closed manifold with non-trivial S^{1}-action. Then the action has no fixed points and the inclusion of any orbit into M induces an injection on the fundamental groups. Moreover, M̃ is det-L^{2}-acyclic and

ρ^{(2)}(M̃) = 0.

The assertion for the L^{2}-torsion in the theorem below is the main result of [106] (see also [107]). Its proof is based on localization techniques.

Theorem 11.22 (L^{2}-torsion and aspherical CW -complexes). Let X be an aspherical finite CW -complex. Suppose that its fundamental group π_{1}(X) contains an elementary amenable infinite normal subgroup H and π_{1}(X) is of det ≥ 1-class. Then X̃ is det-L^{2}-acyclic and

ρ^{(2)}(X̃) = 0.

Remark 11.23 (Homotopy invariance of L^{2}-torsion). Notice that Conjecture 11.14 implies because of Theorem 11.18 (i) the homotopy invariance of the L^{2}-torsion. i.e. for two homotopy equivalent det-L^{2}-acyclic finite CW - complexes X and Y we have ρ^{(2)}(X̃) = ρ^{(2)}(Ỹ).

11.4 Computations of L^{2}-Torsion

Remark 11.24 (Analytic L^{2}-torsion). It is important to know for the following specific calculations that there is an analytic version of L^{2}-torsion in terms of the heat kernel due to Lott [66] and Mathai [86] and that a deep result of Burghelea, Friedlander, Kappeler and McDonald [9] says that the analytic one agrees with the one presented here.

The following result is due to Hess and Schick [51].

Theorem 11.25 (Analytic L^{2}-torsion of hyperbolic manifolds). Let d = 2n+1 be an odd integer. To d one can associate an explicit real number C_{d} > 0 with the following property:

For every closed hyperbolic d-dimensional manifold M we have

ρ^{(2)}(M̃) = (−1)^{n} · C_{d} · vol(M),

where vol(M) is the volume of M.

The existence of a real number C_{d} with ρ^{(2)}(M̃) = (−1)^{n} · C_{d} · vol(M) follows from the version of the Proportionality Principle for L^{2}-Betti numbers (see Theorem 3.7) for L^{2}-torsion (see [80, Theorem 3.183]). The point is that this number C_{d} is given explicitly. For instance C_{3} = \frac{1}{6π} and C_{5} = \frac{31}{45π^{2}} . For each odd d there exists a rational number r_{d} such that C_{d} = π^{−n} · r_{d} holds. The proof of this result is based on calculations involving the heat kernel on hyperbolic space.

Remark 11.26 (L^{2}-torsion of symmetric spaces of non-compact type). More generally, the L^{2}-torsion ρ^{(2)}(M̃) for an aspherical closed manifold M whose universal covering M̃ is a symmetric space is computed by Olbricht [93].

The following result is proved in [84, Theorem 0.6].

Theorem 11.27 (L^{2}-torsion of 3-manifolds). Let M be a compact connected orientable prime 3-manifold with infinite fundamental group such that the boundary of M is empty or a disjoint union of incompressible tori. Suppose that M satisfies Thurston’s Geometrization Conjecture, i.e. there is a geometric toral splitting along disjoint incompressible 2-sided tori in M whose pieces are Seifert manifolds or hyperbolic manifolds. Let M_{1}, M_{2}, …, M_{r} be the hyperbolic pieces. They all have finite volume [90, Theorem B on page 52]. Then M̃ is det-L^{2}-acyclic and

ρ^{(2)}(M̃) = − \frac{1}{6π} · ∑_{i=1}^{r} vol(M_{i}).

In particular, ρ^{(2)}(M̃) is 0 if and only if there are no hyperbolic pieces.

11.5 Some Open Conjectures about L^{2}-Torsion

All the computations and results above give evidence and are compatible with the following conjectures about L^{2}-torsion taken from [80, Theorem 11.3].

Conjecture 11.28 (L^{2}-torsion for aspherical manifolds). If M is an aspherical closed manifold of odd dimension, then M̃ is det-L^{2}-acyclic and

(−1)^{\frac{dim(M)−1}{2}} · ρ^{(2)}(M̃) ≥ 0.

If M is a closed connected Riemannian manifold of odd dimension with negative sectional curvature, then M̃ is det-L^{2}-acyclic and

(−1)^{\frac{dim(M)−1}{2}} · ρ^{(2)}(M̃) > 0.

If M is an aspherical closed manifold whose fundamental group contains an amenable infinite normal subgroup, then M̃ is det-L^{2}-acyclic and

ρ^{(2)}(M̃) = 0.

Consider a closed orientable manifold M of dimension n. Let [M; ℝ] be the image of the fundamental class [M] ∈ H_{n}^{sing}(M; ℤ) under the change of coefficient map H_{n}^{sing}(M; ℤ) → H_{n}^{sing}(M; ℝ). Define the L^{1}-norm on C_{n}^{sing}(M; ℝ) by sending ∑_{i=1}^{s} r_{i} · [σ_{i}: Δ_{n} → M] to ∑_{i=1}^{s} |r_{i}|. It induces a seminorm on H_{n}(M; ℝ). Define the simplicial volume ||M|| ∈ ℝ to be the seminorm of [M; ℝ]. More information about the simplicial volume can be found for instance in [42],

[47] and [57], and in [80, Chapter 14], where also the following conjecture is discussed.

Conjecture 11.29 (Simplicial volume and L^{2}-invariants). Let M be an aspherical closed orientable manifold of dimension ≥ 1. Suppose that its simplicial volume ||M|| vanishes. Then M̃ is det-L^{2}-acyclic and

ρ^{(2)}(M̃) = 0.

The simplicial volume is a special invariant concerning bounded cohomology. The point of this conjecture is that it suggests a connection between bounded cohomology and L^{2}-invariants such as L^{2}-cohomology and L^{2}-torsion.

We have already seen that L^{2}-Betti numbers are up to scaling invariant under measure equivalence. The next conjecture is interesting because it would give a sharper invariant in case all the L^{2}-Betti numbers vanish, namely the vanishing of the L^{2}-torsion.

Conjecture 11.30 (Measure equivalence and L^{2}-torsion). Let G_{i} for i = 0, 1 be a group such that there is a finite CW -model for BG_{i} and EG_{i} is det-L^{2}-acyclic. Suppose that G_{0} and G_{1} are measure equivalent. Then

ρ^{(2)}(EG_{0}; N (G_{0})) = 0 ⇔ ρ^{(2)}(EG_{1}; N (G_{1})) = 0.

11.6 L^{2}-Torsion of Group Automorphisms

In this section we explain that for a group automorphism f: G → G the L^{2}- torsion applied to the (G ⋊_{f} ℤ)-CW -complex E(G ⋊_{f} ℤ) gives an interesting new invariant, provided that G is of det ≥ 1-class and satisfies certain finiteness assumptions. It seems to be worthwhile to investigate it further. The following definition and theorem are taken from [80, Definition 7.26 and Theorem 7.27].

Definition 11.31 (L^{2}-torsion of group automorphisms). Let f: G → G be a group automorphism. Suppose that there is a finite CW -model for BG and G is of det ≥ 1-class. Define the L^{2}-torsion of f by

ρ^{(2)}(f: G → G) := ρ^{(2)}({ \atop B(G ⋊_{f} ℤ)}) ∈ ℝ.

Next we present the basic properties of this invariant. Notice that its behaviour is similar to the Euler characteristic χ(G) := χ(BG).

Theorem 11.32. Suppose that all groups appearing below have finite CW - models for their classifying spaces and are of det ≥ 1-class.

(i) Suppose that G is the amalgamated product G_{1}∗_{G_{0}}G_{2} for subgroups G_{i} ⊆ G and the automorphism f: G → G is the amalgamated product f_{1} ∗_{f_{0}} f_{2} for automorphisms f_{i}: G_{i} → G_{i}. Then

ρ^{(2)}(f) = ρ^{(2)}(f_{1}) + ρ^{(2)}(f_{2}) − ρ^{(2)}(f_{0});

(ii) Let f: G → H and g: H → G be isomorphisms of groups. Then

ρ^{(2)}(f ∘ g) = ρ^{(2)}(g ∘ f).

In particular ρ^{(2)}(f) is invariant under conjugation with automorphisms; (iii) Suppose that the following diagram of groups

1 ⟶ G_{1} ⟶ G_{2} ⟶ G_{3} ⟶ 1
f_{1}↓  ^{f_{2}}↓  ^{id}↓
1 ⟶ G_{1} ⟶ G_{2} ⟶ G_{3} ⟶ 1

commutes, has exact rows and its vertical arrows are automorphisms. Then

ρ^{(2)}(f_{2}) = χ(BG_{3}) · ρ^{(2)}(f_{1});

(iv) Let f: G → G be an automorphism of a group. Then for all integers n ≥ 1

ρ^{(2)}(f^{n}) = n · ρ^{(2)}(f);

(v) Suppose that G contains a subgroup G_{0} of finite index [G : G_{0}]. Let f: G → G be an automorphism with f(G_{0}) = G_{0}. Then

ρ^{(2)}(f) = \frac{1}{[G : G_{0}]} · ρ^{(2)}(f|_{G_{0}});

(vi) We have ρ^{(2)}(f) = 0 if G satisfies one of the following conditions:

(a) All the L^{2}-Betti numbers of G vanish;

(b) G contains an amenable infinite normal subgroup.

Example 11.33 (Automorphisms of surfaces). Using Theorem 11.27 one can compute the L^{2}-torsion of the automorphism π_{1}(f) for an automorphism f: S → S of a compact connected orientable surface, possibly with boundary. Suppose that f is irreducible. Then the following statements are equivalent: i.) f is pseudo-Anosov, ii.) The mapping torus T_{f} has a hyperbolic structure and iii.) ρ^{(2)}(π_{1}(f)) < 0. Moreover, f is periodic if and only if ρ^{(2)}(π_{1}(f)) = 0. (see [80, Subsection 7.4.2]).

The L^{2}-torsion of a Dehn twist is always zero since the associated mapping torus contains no hyperbolic pieces in his Jaco-Shalen-Johannson-Thurston splitting.

Remark 11.34 (Weaker finiteness conditions). The definition of the L^{2}- torsion of a group automorphism above still makes sends and has still most of the properties above, if one weakens the condition that there is a finite model for BG to the assumption that there is a finite model for the classifying space of proper G-actions \underline{E}G = E_{FIN} (G). This is explained in [80, Subsection 7.4.4].

12 Novikov-Shubin Invariants

In this section we briefly discuss Novikov-Shubin invariants. They were originally defined in terms of heat kernels. We will focus on their algebraic definition and aspects.

12.1 Definition of Novikov-Shubin Invariants

Let M be a finitely presented N (G)-module. Choose some exact sequence N (G)^{m} ⟶^{f} N (G)^{n} → M → 0. Let F_{f} be the spectral density function of f (see Definition 11.1). Recall that F_{f} is a monotone increasing right continuous function [0, ∞) → [0, ∞). Define the Novikov-Shubin invariant of M by

α(M) = lim inf_{λ→0^{+}} \frac{ln(F_{f}(λ) − F_{f}(0))}{ln(λ)} ∈ [0, ∞],

provided that F_{f}(λ) > F_{f}(0) holds for all λ > 0. Otherwise, one puts formally

α(M) = ∞^{+}.

It measures how fast F_{f}(λ) approaches F_{f}(0) for λ → 0^{+}. For instance, if F_{f}(λ) = λ^{α} for λ > 0, then α(M) = α. The proof that α(M) is independent of the choice of f is analogous to the proof of [80, Theorem 2.55 (1)].

Definition 12.1 (Novikov-Shubin invariants). Let X be a G-CW -complex of finite type. Define its p-th Novikov-Shubin invariant by

α_{p}(X; N (G)) = α (H_{p−1}^{(2)}(X; N (G))) ∈ [0, ∞] ⨿ {∞^{+}}.

If the group G is clear from the context, we abbreviate α_{p}(X) = α_{p}(X; N (G)).

Notice that H_{p−1}^{(2)}(X; N (G)) is finitely presented since N (G) is semihereditary (see Theorem 1.6) and C_{k}^{(2)}(X) is a finitely generated free N (G)-module for all k ∈ ℤ because X is by assumption of finite type.

Remark 12.2 (Analytic definition of Novikov-Shubin invariants). Novikov-Shubin invariants were originally analytically defined by Novikov and Shubin (see [91], [92]). For a cocompact smooth G-manifold M without boundary and with G-invariant Riemannian metric one can assign to its p-th Laplace operator Δ_{p} a density function F_{Δ_{p}}(λ) = tr_{N (G)}(E_{λ}) for {E_{λ} | λ ∈ [0, ∞)} the spectral family associated to the essentially selfadjoint operator Δ_{p}. Define α_{p}^{Δ}(M; N (G)) ∈ [0, ∞] ⨿ {∞^{+}} by the same expression as appearing in the definition of α(M) above, only replace F_{f} by F_{Δ_{p}}. Then α_{p}^{Δ}(M) agrees with \frac{1}{2} ·min\{α_{p}(K), α_{p+1}(K)\} for any equivariant triangulation K of M. For a proof of this equality see [31] or [80, Section 2.4]. One can define the analytic Novikov-Shubin invariant α_{p}^{Δ}(M; N (G)) also in terms of heat kernels. It measures how fast the function ∫_{F} tr_{ℂ}(e^{−tΔ_{p}}(x, x)) dvol_{x} approaches for t → ∞ its limit

b_{p}^{(2)}(M) = lim_{t→∞} ∫_{F} tr_{ℂ}(e^{−tΔ_{p}}(x, x)) dvol_{x}.

The “thinner” the spectrum of Δ_{p} is at zero, the larger is α_{p}^{Δ}(M; N (G)).

In view of this original analytic definition the result due to Gromov and Shubin [48] that the Novikov-Shubin invariants are homotopy invariants, is rather surprising.

Remark 12.3 (Analogy to finitely generated ℤ-modules). Recall Slogan 1.16 that the group von Neumann algebra N (G) behaves like the ring of integers ℤ, provided one ignores the properties integral domain and Noetherian. Given a finitely generated abelian group M, the ℤ-module M/ tors(M) is finitely generated free, there is a ℤ-isomorphism M ≅ M/ tors(M) ⊕ tors(M) and the rank as abelian group of M is dim_{ℚ}(ℚ ⊗_{ℤ} M) and of tors(M) is 0. In analogy, given a finitely generated N (G)-module M, then PM := M/TM is a finitely generated projectiveN (G)-module, there is an N (G)-isomorphism M ≅ PM ⊕ tors(M) and we get dim_{N (G)}(M) = dim_{U(G)}(U(G) ⊗_{N (G)} M) and dim_{N (G)}(TM) = 0. Define the so called capacity c(M) ∈ [0, ∞] ∪ {0^{−}} of a finitely presented N (G)-module M by

c(M) = \{ \begin{matrix}\frac{1}{α(M)} & if α(M) ∈ (0, ∞); \\ ∞ & if α(M) = 0; \\ 0 & if α(M) = ∞; \\ 0^{−} & if α(M) = ∞^{+}. \end{matrix}

Then the capacity c(M) contains the same information as α(M) and corresponds under the dictionary between ℤ and N (G) to the order of the finite group tors(M). Notice for a finitely presented N (G)-module M that M = 0 is true if and only if both dim_{N (G)}(M) = 0 and c(M) = 0^{−} hold. The capacity is at least subadditive, i.e. for an exact sequence 1 → M_{0} → M_{1} → M_{2} → 0 of finitely presented N (G)-modules we have c(M_{1}) ≤ c(M_{0}) + c(M_{2}) (with the obvious interpretation of + and ≤). In particular we get c(M) ≤ c(N) for an inclusion of finitely presented N (G)-modules M ⊆ N.

Remark 12.4 (Extension to arbitrary N (G)-modules and G-spaces). The algebraic approach presented above has been independently developed in [33] and [74]. The notion of capacity has been extended by Lück-Reich-Schick [82] to so called cofinal-measurable N (G)-modules, i.e. N (G)-modules such that each finitely generated N (G)-submodule is a quotient of a finitely presented N (G)-module with trivial von Neumann dimension. This allows to define Novikov-Shubin invariants for arbitrary G-spaces and also for arbitrary groups G.

12.2 Basic Properties of Novikov-Shubin Invariants

We briefly list some properties of Novikov-Shubin invariants. The proof of the following theorem can be found in [80, Theorem 2.55] and [80, Lemma 13.45].

Theorem 12.5 (Novikov-Shubin invariants).

(i) Homotopy invariance Let f: X → Y be a G-map of free G-CW -complexes of finite type. Suppose that the map H_{p}(f; ℂ): H_{p}(X; ℂ) → H_{p}(Y; ℂ) induced on homology with complex coefficients is an isomorphism for p ≤ d − 1. Then we get

α_{p}(X; N (G)) = α_{p}(Y; N (G)) for p ≤ d.

In particular we get α_{p}(X; N (G)) = α_{p}(Y; N (G)) for all p ≥ 0 if f is a weak homotopy equivalence;

(ii) Poincaré duality Let M be a cocompact free proper G-manifold of dimension n which is orientable. Then α_{p}(M; N (G)) = α_{n+1−p}(M, ∂M; N (G)) for p ≥ 1;

(iii) First Novikov-Shubin invariant Let X be a connected free G-CW -complex of finite type. Then G is finitely generated and

(a) α_{1}(X) is finite if and only if G is infinite and virtually nilpotent. In this case α_{1}(X) is the growth rate of G;

(b) α_{1}(X) is ∞^{+} if and only if G is finite or non-amenable;

(c) α_{1}(X) is ∞ if and only if G is amenable and not virtually nilpotent;

(iv) Restriction to subgroups of finite index Let X be a free G-CW -complex of finite type and H ⊆ G a subgroup of finite index. Then α_{p}(X; N (G)) = α_{p}(res_{G}^{H} X; N (H)) holds for p ≥ 0;

(v) Extensions with finite kernel Let 1 → H → G → Q → 1 be an extension of groups such that H is finite. Let X be a free Q-CW -complex of finite type. Then we get α_{p}(p^{∗}X; N (G)) = α_{p}(X; N (Q)) for all p ≥ 1;

(vi) Induction Let H be a subgroup of G and let X be a free H-CW -complex of finite type. Then α_{p}(G ×_{H} X; N (G)) = α_{p}(X; N (H)) holds for all p ≥ 1.

A product formula and a formula for connected sums can also be found in [80, Theorem 2.55]. If X is a finite G-CW -complex such that b_{p}^{(2)}(X; N (G)) = 0 for p ≥ 0 and α_{p}(X; N (G)) > 0 for p ≥ 1, then X is det-L^{2}-acyclic [80, Theorem 3.93 (7)].

12.3 Computations of Novikov-Shubin Invariants

Example 12.6 (Novikov-Shubin invariants of T̃^{n}). The product formula can be used to show α_{p}(T̃^{n}) = n if 1 ≤ p ≤ n, and α_{p}(T̃^{n}) = ∞^{+} otherwise (see [80, Example 2.59].)

Example 12.7 (Novikov-Shubin invariants for finite groups). If G is finite, then α_{p}(X; N (G)) = ∞^{+} for each p ≥ 1 and G-CW -complex X of finite type. This follows from Example 11.2. This shows that the Novikov-Shubin invariants are interesting only for infinite groups G and have no classical analogue in contrast to L^{2}-Betti numbers and L^{2}-torsion.

Example 12.8 (Novikov-Shubin invariants for G = ℤ). Let X be a free ℤ-CW -complex of finite type. Since ℂ[ℤ] is a principal ideal domain, we get ℂ[ℤ]-isomorphisms

H_{p}(X; ℂ) ≅ ℂ[ℤ]^{n_{p}} ⊕ (⨁_{i_{p}=1}^{s_{p}} ℂ[ℤ]/((z − a_{p,i_{p}})^{r_{p,ip}}))

for a_{p,i_{p}} ∈ ℂ and n_{p}, s_{p}, r_{p,i_{p}} ∈ ℤ with n_{p}, s_{p} ≥ 0 and r_{p,i_{p}} ≥ 1, where z is a fixed generator of ℤ. Then we get from [80, Lemma 2.58]

b_{p}^{(2)}(X; N (ℤ)) = n_{p}.

If s_{p} ≥ 1 and {i_{p} = 1, 2 …, s_{p}, |a_{p,i_{p}}| = 1} ≠ ∅, then

α_{p+1}(X; N (ℤ)) = min \{\frac{1}{r_{p,i_{p}}} | i_{p} = 1, 2 …, s_{p}, |a_{p,i_{p}}| = 1\} ,

and otherwise

α_{p+1}(X; N (ℤ)) = ∞^{+}.

Remark 12.9 (Novikov-Shubin invariants and S^{1}-actions). Under the conditions of Theorem 3.8 and of Theorem 3.9 one can show α_{p}(X̃) ≥ 1 for all p ≥ 1 (see [80, Theorem 2.61 and Theorem 2.63]).

Remark 12.10 (Novikov-Shubin invariants of symmetric spaces of non-compact type). The Novikov-Shubin invariants of symmetric spaces of non-compact type with cocompact free G-action are computed by Olbricht [93, Theorem 1.1], the result is also stated in [80, Section 5.3].

Remark 12.11 (Novikov-Shubin invariants of universal coverings of 3-manifolds). Partial results about the computation of the Novikov-Shubin invariants of universal coverings of compact orientable 3-manifolds can be found in [69] and [80, Theorem 4.2].

12.4 Open Conjectures about Novikov-Shubin invariants

The following conjecture is taken from [69, Conjecture 7.1 on page 56].

Conjecture 12.12. (Positivity and rationality of Novikov-Shubin invariants). Let G be a group. Then for any free G-CW -complex X of finite type its Novikov-Shubin invariants α_{p}(X) are positive rational numbers unless they are ∞ or ∞^{+}.

This conjecture is equivalent to the statement that for every finitely presented ℤG-module M the Novikov-Shubin invariant of N (G)⊗_{ℤG}M is a positive rational number, ∞ or ∞^{+}.

Here is some evidence for Conjecture 12.12. Unfortunately, all the evidence comes from computations, no convincing conceptual reason is known. Conjecture 12.12 is true for G = ℤ by the explicit computation appearing in Example 12.8. Conjecture 12.12 is true for virtually abelian G by [66, Proposition 39 on page 494]. Conjecture 12.12 is also true for a free group G. Details of the proof appear in the Ph.D. thesis of Roman Sauer [100] following ideas of Voiculescu. The essential ingredients are non-commutative power series and the question whether they are algebraic or rational. All the computations mentioned above are compatible and give evidence for Conjecture 12.12.

Conjecture 12.13 (Zero-in-the-spectrum Conjecture). Let G be a group such that BG has a closed manifold as model. Then there is p ≥ 0 with H_{p}^{G}(EG; N (G)) ≠ 0.

Remark 12.14 (Original zero-in-the-spectrum Conjecture). The original zero-in-the-spectrum Conjecture, which appears for the first time in Gromov’s article [43, page 120], says the following: Let M̃ be a complete Riemannian manifold. Suppose that M̃ is the universal covering of an aspherical closed Riemannian manifold M (with the Riemannian metric coming from M). Then for some p ≥ 0 zero is in the spectrum of the minimal closure

(Δ_{p})_{min} : dom ((Δ_{p})_{min}) ⊆ L^{2}Ω^{p}(M̃) → L^{2}Ω^{p}(M̃)

of the Laplacian acting on smooth p-forms on M̃.

It follows from [80, Lemma 12.3] that this formulation is equivalent to the homological algebraic formulation appearing in Conjecture 12.13.

Remark 12.15 (Status of the zero-in-the-spectrum Conjecture). The zero-in-the-spectrum Conjecture is true for G if there is a closed manifold model for BG which is Kähler hyperbolic [45], or whose universal covering is hyper-Euclidean [43] or is uniformly contractible with finite asymptotic dimension [110]. The zero-in-the-spectrum Conjecture is true for G if the strong Novikov Conjecture holds for G [67]. More information about zero-in-the-spectrum Conjecture can be found for instance in [67] and [80, Section 12].

Remark 12.16 (Variations of the zero-in-the-spectrum Conjecture). One may ask whether one can weaken the condition in Conjecture 12.13 that BG has a closed manifold model to the condition that there is a finite CW -complex model for BG. This would rule out Poincaré duality from the picture. Or one could only require that BG is of finite type. Without any finiteness conditions on G Conjecture 12.13 is not true in general. For instance H_{p}^{G}(EG; N (G)) = 0 holds for all p ≥ 0 if G is ∏_{i=1}^{∞} ℤ ∗ ℤ.

The condition aspherical cannot be dropped. Farber and Weinberger [35] proved the existence of a closed non-aspherical manifold M with fundamental group π a product of three copies of ℤ∗ℤ such that H_{p}^{π}(M̃; N (π)) vanishes for all p ≥ 0. Later Higson-Roe-Schick [52] proved that one can find for every finitely presented group π, for which H_{p}^{π}(Eπ; N (π)) = 0 holds for p = 0, 1, 2, a closed manifold M with π as fundamental group such that H_{p}^{π}(M̃; N (π)) vanishes for all p ≥ 0.

Remark 12.17 (Novikov-Shubin invariants and quasi-isometry). Since α_{1}(ℤ^{n}) = n for n ≥ 1, the Novikov-Shubin invariants are not invariant under measure equivalence. It is not known whether they are invariant under quasiisometry. At least it is known that two quasi-isometric amenable groups G_{1} and G_{2} which possess finite models for BG_{1} and BG_{2} have the same Novikov-Shubin invariants [100]. Compare also Theorem 8.2, Remark 8.5 and Conjecture 11.30.

13 A Combinatorial Approach to L^{2}-Invariants

In this section we want to give a more combinatorial approach to the L^{2}- invariants such as L^{2}-Betti numbers, Novikov-Shubin invariants and L^{2}-torsion. The point is that it is in general very hard to compute the spectral density function of an N (G)-map f: N (G)^{m} → N (G)^{n}. However in the geometric situation these morphisms are induced by matrices over the integral group ring ℤG. We want to exploit this information to get an algorithm which produces a sequence of rational numbers converging to the L^{2}-Betti number or the L^{2}-torsion in question.

Let A ∈ M(m, n; ℂG) be an (m, n)-matrix over ℂG. It induces by right multiplication an N (G)-homomorphism r_{A}: N (G)^{m} → N (G)^{n}. We define an involution of rings on ℂG by sending ∑_{g∈G} λ_{g} · g to ∑_{g∈G} \overline{λ_{g}} · g^{−1}, where \overline{λ_{g}} is the complex conjugate of λ_{g}. Denote by A^{∗} the (m, n)-matrix obtained from A by transposing and applying the involution above to each entry. Define the ℂGtrace of an element u = ∑_{g∈G} λ_{g} ·g ∈ ℂG by the complex number tr_{ℂG}(u) := λ_{e} for e the unit element in G. This extends to a trace of square (n, n)-matrices A over ℂG by

tr_{ℂG}(A) := ∑_{i=1}^{n} tr_{ℂG}(a_{i,i}) ∈ ℂ.  (13.1)

We get directly from the definitions that the ℂG-trace tr_{ℂG}(u) for u ∈ ℂG agrees with the von Neumann trace tr_{N (G)}(u) introduced in Definition 1.8.

Let A ∈ M(m, n; ℂG) be an (m, n)-matrix over ℂG. In the sequel let K be any positive real number satisfying K ≥ ||r_{A}^{(2)}||, where ||r_{A}^{(2)}|| is the operator norm of the bounded G-equivariant operator r_{A}^{(2)} : l^{2}(G)^{m} → l^{2}(G)^{n} induced by right multiplication with A. For u = ∑_{g∈G} λ_{g} · g ∈ ℂG define ||u||_{1} by ∑_{g∈G} |λ_{g}|. Then a possible choice for K is

K = \sqrt{(2n − 1)m} · max \{||a_{i,j}||_{1} | 1 ≤ i ≤ n, 1 ≤ j ≤ m\} .

Definition 13.2. The characteristic sequence of a matrix A ∈ M(m, n; ℂG) and a non-negative real number K satisfying K ≥ ||r_{A}^{(2)}|| is the sequence of real numbers given by

c(A, K)_{p} := tr_{ℂG} ((1 − K^{−2} · AA^{∗})^{p}) .

We have defined dim_{N (G)}(ker(r_{A})) in Definition 1.12 and det_{N (G)}(r_{A}) in Definition 11.4. The proof of the following result can be found in [73] or [80, Theorem 3.172].

Theorem 13.3. (Combinatorial computation of L^{2}-invariants). Let A ∈ M(m, n; ℂG) be an (m, n)-matrix over ℂG. Let K be a positive real number satisfying K ≥ ||r_{A}^{(2)}||. Then:

(i) Monotony The characteristic sequence (c(A, K)_{p})_{p≥1} is a monotone decreasing sequence of non-negative real numbers;

(ii) Dimension of the kernel

We have

dim_{N (G)}(ker(r_{A})) = lim_{p→∞} c(A, K)_{p};

(iii) Novikov-Shubin invariants of the cokernel Define β(A) ∈ [0, ∞] by

β(A) := sup {β ∈ [0, ∞) | lim_{p→∞} p^{β} · (c(A, K)_{p} − dim_{N (G)}(ker(r_{A}))) = 0} .

If α(coker(r_{A})) < ∞, then α(coker(r_{A})) ≤ β(A) and if α(coker(r_{A})) ∈ {∞, ∞^{+}}, then β(A) = ∞;

(iv) Fuglede-Kadison determinant The sum of positive real numbers

∑_{p=1}^{∞} \frac{1}{p} · (c(A, K)_{p} − dim_{N (G)}(ker(r_{A})))

converges if and only if r_{A} is of determinant class and in this case

ln(det(r_{A})) = (n − dim_{N (G)}(ker(r_{A}))) · ln(K)

− \frac{1}{2} · ∑_{p=1}^{∞} \frac{1}{p} · (c(A, K)_{p} − dim_{N (G)}(ker(r_{A})));

(v) Speed of convergence Suppose α(coker(r_{A})) > 0. Then r_{A} is of determinant class. Given a real number α satisfying 0 < α < α(coker(r_{A})), there is a real number C such that we have for all L ≥ 1

0 ≤ c(A, K)_{L} − dim_{N (G)}(ker(r_{A})) ≤ \frac{C}{L^{α}}

and

0 ≤ − ln(det(r_{A})) + (n − dim_{N (G)}(ker(r_{A}))) · ln(K)

−\frac{1}{2} · ∑_{p=1}^{L} \frac{1}{p} · (c(A, K)_{p} − dim_{N (G)}(ker(r_{A}))) ≤ \frac{C}{L^{α}} .

Remark 13.4 (Vanishing of L^{2}-Betti numbers and the Atiyah Conjecture). Suppose that the Atiyah Conjecture 4.1 is satisfied for (G, d, ℂ). If we want to show the vanishing of dim_{N (G)}(ker(r_{A})), it suffices to show that for some p ≥ 0 we have c(A, K)_{p} < \frac{1}{d}. It is possible that a computer program spits out such a value after a reasonable amount of calculation time.

14 Miscellaneous

The analytic aspects of L^{2}-invariants are also very interesting. We have already mentioned that L^{2}-Betti numbers were originally defined by Atiyah [2] in context with his L^{2}-index theorem. Other L^{2}-invariants are the L^{2}-Eta-invariant and the L^{2}-Rho-invariant (see Cheeger-Gromov [13], [14]). The L^{2}-Eta-invariant appears in the L^{2}-index theorem for manifolds with boundary due to Ramachandran [97]. These index theorems have generalizations to a C^{∗}-setting due to Miščenko-Fomenko [89]. There is also an L^{2}-version of the signature. It plays an important role in the work of Cochran, Orr and Teichner [16] who show that there are non-slice knots in 3-space whose Casson-Gordon invariants are all trivial. Chang and Weinberger [11] show using L^{2}-invariants that for a closed oriented smooth manifold M of dimension 4k + 3 for k ≥ 1 whose fundamental group has torsion there are infinitely many smooth manifolds which are homotopy equivalent to M (and even simply and tangentially homotopy equivalent to M) but not homeomorphic to M. The L^{2}-cohomology has also been investigated for complete non-necessarily compact Riemannian manifolds without a group action. For instance algebraic and arithmetic varieties have been studied. In particular, the Cheeger-Goresky-MacPherson Conjecture [12] and the Zucker Conjecture [111] have created a lot of activity. They link the L^{2}-cohomology of the regular part with the intersection homology of an algebraic variety.

Finally we mention other survey articles which deal with L^{2}-invariants: [30], [39], [46, Section 8], [67], [75], [78], [79], [87] and [96].

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Notation

BG, 12 b_{p}^{(2)}(G), 12 b_{p}^{(2)}(\underline{G}), 57 b_{p}^{(2)}(G ↷ X), 58 b_{p}^{(2)}(X; N (G)), 12 ℂ\underline{G}, 55 C_{∗}^{(2)}(X), 19 det_{N (G)}(f), 65 dim_{N (G)}(M), 9 dim_{N (G)}^{u}(P), 47 dim_{U(G)}(M), 30 EG, 12 F_{g}^{d}, 23 F_{f}, 64 f-rk(M), 24 G_{f}, 47 G_{0}(R), 46 H_{p}^{G}(X; N (G)), 12 H_{p}^{(2)}(X; l^{2}(G)), 19 h^{(2)}(G), 19 h^{(2)}(X; N (G)), 19 i_{∗}M, 11 I(X), 19 K_{0}(R), 46 l^{2}(G), 4 L^{2}(T^{n}), 5 L^{∞}(T^{n}), 5 m(X; G), 19 \overline{M}, 8 ||M||, 72 R ∗_{c} G, 56 PM, 8 TM, 8 T_{f}, 27 tr_{ℂG}(A), 80 tr_{N (G)}, 7 tr_{N (G)}^{u}, 46 vol(M), 71 Wh(G), 51 X̃, 12 α(M), 75 α_{p}^{Δ}(M; N (G)), 75

χ^{(2)}(G), 19 ; χ^{(2)}(X; N (G)), 19 ; χ_{virt}(X), 22 ; A, 5 ; B(H), 4 ; B_{d}, 40 ; C, 34 ; D, 34 ; EAM, 34 ; FIN (G), 35 ; N (G), 5 ; N (i), 5 ; R(G ↷ X), 55 ; U(G), 30 ; [0, ∞], 8 ; \frac{1}{|FIN (G)|}ℤ, 35

Index

action essentially free, 55 standard, 55

aspherical, 24

capacity, 76 characteristic sequence, 81 classifying space

for free proper G-actions, 12 of a group, 12

closed under directed unions, 34 extensions, 34

closure of a submodule, 8 cocompact, 14 Conjecture

Amenability and dimension-flatness

of N (G) over ℂG, 37 Amenability and the regular rep-

resentation in G-theory, 51 Approximation Conjecture, 62 Atiyah Conjecture, 29 Atiyah Conjecture for arbitrary

groups, 35 Determinant Conjecture, 67 Flatness of N (G) over ℂG, 37 Hopf Conjecture, 59 Kaplanski Conjecture, 33 L^{2}-torsion for aspherical mani-

folds, 72 Measure equivalence and L^{2}-torsion,

73 Positivity of Novikov-Shubin in-

variants, 78 Ring theoretic version of the Atiyah

Conjecture, 31 Simplicial volume and L^{2}-invariants,

73 Singer Conjecture, 59 Triviality of the map induced

by the Fuglede-Kadison determinant on Wh(G), 68

crossed product ring, 56 deficiency, 43 det ≥ 1-class, 67 det-L^{2}-acyclic, 68 dimension

von Neumann dimension of a finitely generated projective N (G)-module, 7 von Neumann dimension of an N (G)-module, 9 discrete measurable groupoid, 54 discrete measured groupoid, 54

restricted, 54 division closure, 31

essentially free action, 55

Euler characteristic L^{2}-Euler characteristic, 19 orbifold Euler characteristic, 22 virtual Euler characteristic, 22

exceptional prime 3-manifold, 24

factor, 48

faithfully flat functor, 11

flag complex, 60

Fuglede-Kadison determinant

of N (G)-maps N (G)^{m} → N (G)^{n},

65 of endomorphisms of finitely gen-

erated projective N (G)-modules, 67

function

essentially bounded, 5 fundamental rank of a symmetric space,

24 fundamental square of ring exten-

sions, 30

G-CW -complex, 13

finite, 14

finite dimensional, 14 of dimension ≤ n, 14 of finite type, 14 proper, 14 Grothendieck group of finitely gen-

erated modules, 46 group

amenable, 34 det ≥ 1-class, 67 elementary amenable, 34 lamplighter group, 35 measure equivalent, 52 quasi-isometric, 53 Thompson’s group, 41 virtually cyclic, 37

group von Neumann algebra, 5 groupoid, 54

discrete measurable, 54 discrete measured, 54

groupoid ring, 55

Hilbert N (G)-module

finitely generated, 18 Hilbert algebra, 56

index of a coupling triple, 52 of a weak orbit equivalence, 55

integral domain, 6 irreducible 3-manifold, 23

Kähler hyperbolic manifold, 61

L^{2}-Betti number analytic, 18 for arbitrary G-spaces, 12 of a discrete measured groupoid,

57 of a group, 12 of a standard action, 58

L^{2}-Euler characteristic, 19 L^{2}-torsion, 68

of a group automorphism, 73 lamplighter group, 35

manifold exceptional prime 3-manifold, 24 irreducible 3-manifold, 23 Kähler hyperbolic manifold, 61 prime 3-manifold, 23

mapping torus, 27 Markov operator, 35 measurable space, 52 measure coupling, 52 measure equivalent groups, 52

normalized measure, 54 Novikov-Shubin invariant

p-th Novikov-Shubin invariant of a G-CW -complex of finite type, 75

of a finitely generated N (G)- module, 75

orbit equivalence of standard actions,

55 orbit equivalence relation, 55 orbit equivalent standard actions, 55

Polish Space, 52 prime 3-manifold, 23 projective class group, 46

quasi-isometric groups, 53

rational closure, 31

ring crossed product, 56 division closed, 31 groupoid ring, 55 integral domain, 6 rationally closed, 31 semihereditary, 6 von Neumann regular, 30

semihereditary ring, 6 simplicial volume, 72 singular homology with coefficients,

12 skeleton, 14 space

measurable, 52 Polish, 52 standard Borel, 52 spectral density function, 64 standard action, 55

orbit equivalence of, 55 orbit equivalent, 55 weak orbit equivalence of, 55 weakly orbit equivalent, 55

standard Borel space, 52

Theorem

L^{2}-Betti numbers and Betti num-

bers of groups, 38 L^{2}-Betti numbers and homol-

ogy in the amenable case, 36

L^{2}-torsion on aspherical closed

S^{1}-manifolds, 71 Analytic L^{2}-torsion of hyperbolic

manifolds, 71 Cellular L^{2}-torsion for univer-

sal coverings, 69 Combinatorial computation of

L^{2}-invariants, 81 Counterexample to the Atiyah

Conjecture for arbitrary groups, 36

Detecting G_{0}(ℂG) by K_{0}(N (G))

for amenable groups, 49 Detecting elements in Wh(G),

51 Dimension and colimits, 10 Dimension function for arbitrary

N (G)-modules, 8 Dimension function for arbitrary

U(G)-modules, 30 Dimension-flatness of N (G) over

ℂG for amenable G, 36 Fuglede-Kadison determinant, 66 Induction and dimension, 11 K_{0} of finite von Neumann alge-

bras, 48 L^{2}-Betti number and fibrations,

28 L^{2}-Betti numbers and Novikov-

Shubin invariants of Kähler hyperbolic manifolds, 62

L^{2}-Betti numbers and S^{1}-actions,

26 L^{2}-Betti numbers for arbitrary

spaces, 14 L^{2}-Betti numbers of 3-manifolds,

24 L^{2}-Betti numbers of symmet-

ric spaces of non-compact type, 25

L^{2}-Euler characteristic, 20 L^{2}-torsion and aspherical CW -

complexes, 71 L^{2}-torsion and S^{1}-actions, 70 L^{2}-torsion of 3-manifolds, 72 Lattices of positive deficiency,

46 Linnell’s Theorem, 34 Measure equivalence and L^{2}-Betti

numbers, 52 Novikov-Shubin invariants, 76 Proportionality principle for L^{2}-

Betti numbers, 25 Reformulations of the Atiyah Con-

jecture, 29 Reformulations of the Atiyah Con-

jecture for F = ℚ, 29 Status of the Approximation Con-

jecture, 64 Status of the Determinant Con-

jecture, 68 The Atiyah and the Kaplanski

Conjecture, 33 The universal trace, 46 Vanishing of L^{2}-Betti numbers

of mapping tori, 27 Von Neumann Algebras are semi-

hereditary, 6 trace

center valued, 46 over the complex group ring, 80 universal, 46 von Neumann trace, 7

universal G-principal bundle, 12

von Neumann algebra, 5

of a group, 5 von Neumann dimension

center valued, 47 of a finitely generated Hilbert N (G)-module, 18 of a finitely generated projective N (G)-module, 7

of an N (G)-module, 9 von Neumann regular ring, 30 von Neumann trace, 7

weak orbit equivalence of standard

actions, 55 weakly orbit equivalent standard ac-

tions, 55 Whitehead group, 51