“Dimension theory of arbitrary modules over finite von Neumann algebras and L^{2}-Betti numbers II: Applications to Grothendieck

groups, L^{2}-Euler characteristics and Burnside groups” by Wolfgang Lück

The paper is dedicated to Martin Kneser on the occasion of his seventieth birthday

Abstract

We continue the study of the generalized dimension function. We detect elements in the Grothendieck group G_{0}(ℂΓ) of finitely generated ℂΓ-modules, provided that Γ is amenable. We investigate the class of groups for which the zero-th and first L^{2}-Betti numbers resp. all L^{2}-Betti numbers vanish. We study L^{2}-Euler characteristics and introduce for a discrete group Γ its Burnside group extending the classical notions of Burnside ring and Burnside ring congruences for finite Γ.

Key words: Dimension functions for finite von Neumann algebras, L^{2}-Betti numbers, amenable groups, Grothendieck groups, Burnside groups.

AMS-classification numbers: 55T99

Introduction

In part one of this paper [21] we have defined for a finite von Neumann algebra A and an arbitrary module over A (just viewed as a ring), its dimension

dim(M) ∈ [0, ∞].  (0.1)

It extends the classical notion of von Neumann dimension which is a priori defined for finitely generated Hilbert A-modules and thus for finitely generated projective A-modules. This dimension inherits all the useful properties such as Additivity, Cofinality and Continuity and is uniquely characterized by these properties. This allows to define for any topological space with action of a discrete group Γ its p-th L^{2}-Betti number

b_{p}^{(2)}(X; N (Γ)) := dim(H_{p}^{Γ}(X; N (Γ))  (0.2)

as the dimension of the homology of the N (Γ)-chain complex N (Γ) ⊗_{ℤΓ} C_{∗}^{sing}(X), where C_{∗}^{sing}(X) is the singular chain complex of X and N (Γ) is the von Neumann algebra associated to Γ. For a group Γ we define

b_{p}^{(2)}(Γ) = b_{p}(EΓ; N (Γ)),  (0.3)

where EΓ ⟶ BΓ is the universal Γ-principal bundle. If X is the universal covering of a closed Riemannian manifold, this agrees with the definition of the p-th-L^{2}-Betti number of Atiyah [1] in terms of the heat kernel on X

b_{p}^{(2)}(X) = lim_{t→∞} ∫_{F} tr_{ℝ} (e^{−tΔ_{p}}(x, x)) dvol_{x}  ∈ [0, ∞),  (0.4)

where F is a fundamental domain for the Γ-action. We have shown in [21] that for any infinite amenable group Γ, any ℂΓ-module M and any Γ-space X

dim (Tor_{p}^{ℂΓ}(M, N (Γ))) = 0  for p ≥ 1,  (0.5)
b_{p}^{(2)}(X; N (Γ)) = dim (N (Γ) ⊗_{ℤΓ} H_{p}^{sing}(X));  (0.6)
b_{p}^{(2)}(Γ) = 0  for p ≥ 0,  (0.7)

holds. In particular b_{p}^{(2)}(X; N (Γ)) depends only on the ℤΓ-module given by the p-th singular homology H_{p}^{sing}(X) and 0.7 is just the theorem of Cheeger and Gromov [6, Theorem 0.2 on page 191]. The necessary ingredients of these results for the present paper will be reviewed in Section 1.

Equation 0.5 plays a crucial role in detecting non-trivial elements in the Grothendieck group G_{0}(ℂΓ) of finitely generated ℂΓ-modules for amenable groups Γ which will be investigated in Section 2. We will construct for amenable Γ a map

G_{0}(ℂΓ) ⊗_{ℤ} ℂ ⟶ class(Γ)_{cf},  (0.8)

where class(Γ)_{cf} is the complex vector space of functions from the set con(Γ)_{cf} of finite conjugacy classes (γ) of elements in Γ to ℂ (Lemma 2.3 and Theorem 2.12). This map is related to the Hattori-Stallings rank and the universal center-valued trace and dimension of N (Γ) (Theorem 2.12). In particular we will show that the class of ℂΓ in G_{0}(ℂΓ) generates an infinite cyclic subgroup in G_{0}(ℂΓ) if Γ is amenable and is trivial if Γ contains a free group of rank 2 as subgroup (Remark 2.23).

We will investigate for d = 0, 1, … and d = ∞ the class B_{d} of groups Γ for which b_{p}(EΓ; N (Γ)) = 0 for p ≤ d (Theorem 3.3) and discuss applications in Section 3 (Theorem 3.2).

We analyse L^{2}-Euler characteristics and the Burnside group in Section 4 generalizing the classical notions of Burnside ring, Burnside ring congruences and equivariant Euler characteristic for finite groups to infinite groups (Theorem 4.4, Lemma 4.11, Lemma 4.13, Remark 4.14 and Lemma 4.17). In particular the L^{2}-Euler characteristic extends the notion of virtual Euler characteristic of a group to a larger class of groups and we get some vanishing results (Corollary 4.5).

In Section 5 we analyse the possible values of the L^{2}-Betti numbers (Theorem 5.2). If there is no bound on the orders of finite subgroups of Γ, then any non-negative real number can be realized as b_{p}^{(2)}(X; N (Γ)) for p ≥ 3 and a free Γ-CW -complex X. Otherwise we show for the least common multiple d of the orders of finite subgroups that d · b_{p}^{(2)}(X; N (Γ)) is an integer or infinite for any Γ-space X if this holds for any finite free Γ-CW -complex Y (Theorem 5.2). The last conditions hold for instance for elementary amenable groups and free groups by Linnell [14].

The paper is organized as follows:

0. Introduction
1. Review of the generalized dimension function
2. Dimension functions and G_{0}(ℂΓ)
3. Groups with vanishing L^{2}-Betti numbers
4. L^{2}-Euler characteristics and the Burnside group
5. Values of L^{2}-Betti numbers

References

1. Review of the generalized dimension function

We review some of the notions and results of the dimension function introduced in [21] for a finite von Neumann algebra A as far as needed here in order to keep this paper rather self-contained. Recall that the dual module M^{∗} of a left A-module is the left A-module hom_{A}(M, A), where the A-multiplication is given by (af)(x) = f(x)a^{∗} for f ∈ M^{∗}, x ∈ M and a ∈ A. Let K be a A-submodule of the A-module M. Define the closure of K in M to be the A-submodule of M

\overline{K} := \{x ∈ M | f(x) = 0 for all f ∈ M^{∗} with K ⊂ ker(f)\}.  (1.1)

For a finitely generated A-module M define the A-submodule TM and the A-quotient module PM by:

TM := {x ∈ M | f(x) = 0 for all f ∈ M^{∗}};  (1.2)
PM := M/TM.  (1.3)

The notion of TM and PM corresponds in [12] to the torsion part and the projective part.

Theorem 1.4 1. A is semi-hereditary, i.e. any finitely generated submodule of a projective module is projective;

2. If K ⊂ M is a submodule of the finitely generated A-module M, then M/\overline{K} is finitely generated and projective and \overline{K} is a direct summand in M;

3. If M is a finitely generated A-module, then PM is finitely generated projective and

M ≅ PM ⊕ TM;

4. The dimension dim has the following properties:

(a) Continuity

If K ⊂ M is a submodule of the finitely generated A-module M, then:

dim(K) = dim(\overline{K});

(b) 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(M) = sup{dim(M_{i}) | i ∈ I};

(c) Additivity

If 0 ⟶ M_{0} ⟶^{i} M_{1} ⟶^{p} M_{2} ⟶ 0 is an exact sequence of A-modules, then:

dim(M_{1}) = dim(M_{0}) + dim(M_{2}),

where r + s for r, s ∈ [0, ∞] is the ordinary sum of two real numbers if both r and s are not ∞ and is ∞ otherwise;

(d) Extension Property If M is finitely generated projective, then dim(M) is the von Neumann dimension of the finitely generated Hilbert N (Γ)-module associated to M (see [21, Definition 1.6 and Theorem 1.8];

(e) If M is a finitely generated A-module, then:

dim(M) = dim(PM);
dim(TM) = 0;

(f) The dimension dim is uniquely determined by Continuity, Cofinality, Additivity and the Extension Property.

Notice that in view of Theorem 1.4 there are some similiarities between the ring ℤ of the integers and the ring A given by a finite von Neumann algebra. If one substitutes in the statements of Theorem 1.4 A by ℤ and requires in the Extension Property that dim(M) for a finitely generated abelian group is the usual rank, then all statements remain true and dim(M) becomes the dimension of the rational vector space M ⊗_{ℤ} ℚ. Moreover, TM of a finitely generated ℤ-module is just its torsion submodule in the ordinary sense. However, there are two important differences. A finite von Neumann algebra is in general not Noetherian and hence harder to study than the Noetherian ring ℤ. On the other hand the dimension of a finitely generated projective A-module can be an arbitrary small positive real number and hence the dimension of a countable direct sum of non-trivial finitely generated projective A-modules can be a finite number what can never happen over ℤ (see also [21, Remark 2.14]).

2. Dimension functions and G_{0}(ℂΓ)

Let G_{0}(ℂΓ) be the abelian group which has as set of generators the isomorphism classes of finitely generated (not necessarily projective) ℂΓ-modules and has for each exact sequence of finitely generated ℂΓ-modules 0 ⟶ M_{0} ⟶ M_{1} ⟶ M_{2} ⟶ 0 the relation [M_{0}] − [M_{1}] + [M_{2}] = 0. Given a finitely generated ℂΓ-module M, the N (Γ)-module N (Γ) ⊗_{ℂΓ} M is a finitely generated N (Γ)-module. We have defined TN (Γ) ⊗_{ℂΓ} M and PN (Γ) ⊗_{ℂΓ} M in 1.2 and 1.3. Recall from Theorem 1.4.3 that PN (Γ) ⊗_{ℂΓ} M is a finitely generated projective N (Γ)-module. Define maps

i : K_{0}(ℂΓ) ⟶ G_{0}(ℂΓ) [P] ↦ [P];  (2.1)
k : K_{0}(ℂΓ) ⟶ K_{0}(N (Γ)) [P] ↦ [N (Γ) ⊗_{ℂΓ} P].  (2.2)

Lemma 2.3 If Γ is amenable, the map

j : G_{0}(ℂΓ) ⟶ K_{0}(N (Γ)) [M] ↦ [PN (Γ) ⊗_{ℂΓ} M]

is a well-defined homomorphism. The composition j ∘ i agrees with k for the maps i and k defined in 2.1 and 2.2 above.

\underline{\underline{Proof}} : If 0 ⟶ M_{0} ⟶^{i}M_{1} ⟶^{p}M_{2} ⟶ 0 is an exact sequence of finitely generated ℂΓmodules we have to check in K_{0}(N (Γ))

[PN (Γ) ⊗_{ℂΓ} M_{0}] − [PN (Γ) ⊗_{ℂΓ} M_{1}] + [PN (Γ) ⊗_{ℂΓ} M_{2}] = 0.

Consider the induced sequence PN (Γ) ⊗_{ℂΓ} M_{0} ⟶^{\overline{i}}PN (Γ) ⊗_{ℂΓ} M_{1} ⟶^{\overline{p}}PN (Γ) ⊗_{ℂΓ} M_{2}. Obviously \overline{p} is surjective as p is surjective. We conclude from Theorem 1.4.1 that ker(\overline{i}) and ker(\overline{p}) are a finitely generated projective N (Γ)-modules. Theorem 1.4.4 and 0.5 imply

dim_{N (Γ)} (ker(\overline{i})) = 0;
dim_{N (Γ)} (im(\overline{i})) = dim_{N (Γ)} (ker(\overline{p})) .

Notice that a finitely generated projective A-module is trivial if and only if its dimension is zero. We conclude from Theorem 1.4 that \overline{i} : PN (Γ) ⊗_{ℂΓ} M_{0} ⟶ ker(\overline{p}) is a weak isomorphism, i.e. its kernel is trivial and \overline{im(\overline{i})} = ker(\overline{p}). Since the functor ν of [21, Theorem 1.8] respects weak exactness and the Polar Decomposition Theorem applied to a weak isomorphism has an isomorphism as unitary part, PN (Γ) ⊗_{ℂΓ} M_{0} and ker(\overline{p}) are isomorphic as N (Γ)-modules. Since ker(\overline{p}) ⊕ PN (Γ) ⊗_{ℂΓ} M_{2} and PN (Γ) ⊗_{ℂΓ} M_{1} are isomorphic, Lemma 2.3 follows.

If we regard hom_{ℂΓ}(M, N (Γ)) as left N (Γ)-module by (af)(x) = f(x) · a^{∗} for a ∈ N (Γ), f ∈ hom_{ℂΓ}(M, N (Γ)) and x ∈ M, we obtain isomorphisms of N (Γ)-modules

hom_{ℂΓ}(M, N (Γ)) ⟶^{≅} (N (Γ) ⊗_{ℂΓ} M)^{∗};
(PN (Γ) ⊗_{ℂΓ} M)^{∗} ⟶^{≅} (N (Γ) ⊗_{ℂΓ} M)^{∗} .

Since for a finitely generated projective N (Γ)-module P its dual P^{∗} is isomorphic to P, we conclude for a finitely generated ℂΓ-module M

j([M]) = [hom_{ℂΓ}(M, N (Γ))] .  (2.4)

For a finitely generated projective N (Γ)-module P let

dim_{N (Γ)}^{u}(P) ∈ cent(N (Γ))  (2.5)

be its center-valued von Neumann dimension which is given in terms of the universal centervalued trace tr_{N (Γ)}^{u} [13, Theorem 8.2.8 on page 517, Proposition 8.3.10 on page 525 and Theorem 8.4.3. on page 532], [19, section 3]. The center-valued von Neumann dimension is additive under direct sums and two finitely generated projective N (Γ)-modules P and Q are isomorphic if and only if dim_{N (Γ)}^{u}(P) = dim_{N (Γ)}^{u}(Q). We obtain an injection

dim_{N (Γ)}^{u} : K_{0}(N (Γ)) → cent(N (Γ))^{+} = {a ∈ cent(N (Γ)) | a = bb^{∗} for b ∈ N (Γ))}, (2.6)

which is an isomorphism if N (Γ) is of type II, for instance if Γ is finitely generated and does not contain an abelian subgroup of finite index ([19, Corollary 3.2 and Lemma 3.3]).

Next we investigate the relationship between K_{0}(ℂΓ) and G_{0}(ℂΓ) and between dim_{N (Γ)}^{u} and the Hattori-Stallings rank. Let con(Γ) be the set of conjugacy classes of elements in Γ. Let con(Γ)_{f} be the subset of con(Γ) of conjugacy classes (γ) for which each representative γ has finite order. Let con(Γ)_{cf} be the subset of con(Γ) of conjugacy classes (γ) which contain only finitely many elements. We denote by class_{0}(Γ) and class_{0}(Γ)_{f} respectively the complex vector space with the set con(Γ) and con(Γ)_{f} respectively as basis. We denote by class(Γ) and class(Γ)_{cf} respectively the complex vector space of functions from the set con(Γ) and con(Γ)_{f} respectively to ℂ. Notice that class_{0}(Γ) is the complex vector space of class functions from Γ to ℂ with finite support. Define the universal ℂΓ-trace of ∑_{γ∈Γ} λ_{γ}γ ∈ ℂΓ by

tr_{ℂΓ}^{u} (∑_{γ∈Γ} λ_{γ}γ) := ∑_{γ∈Γ} λ_{γ} · (γ)  ∈ class_{0}(Γ),  (2.7)

This extends to square matrices in the usual way

tr_{ℂΓ}^{u} : M(n, n, ℂΓ) ⟶ class_{0}(Γ) A ↦ ∑_{i=1}^{n} tr_{ℂΓ}^{u}(a_{i,i}).  (2.8)

Let P be a finitely generated projective ℂΓ-module. Define its Hattori-Stallings rank by

HS(P) := tr_{ℂΓ}^{u}(A)  ∈ class_{0}(Γ),  (2.9)

where A is any element in M(n, n, ℂΓ) with A^{2} = A such that the image of the map ℂΓ^{n} ⟶ ℂΓ^{n} given by right multiplication with A is ℂΓ-isomorphic to P. This definition is independent of the choice of A. The Hattori-Stallings rank defines a homomorphism

HS : K_{0}(ℂΓ) ⟶ class_{0}(Γ) [P] ↦ HS(P).  (2.10)

Define a homomorphism

ϕ : cent(N (Γ)) ⟶ class(Γ)_{cf}  (2.11)

by assigning to u ∈ cent(N (Γ))

ϕ(u) : con(Γ)_{cf} ⟶ ℂ (δ) ↦ tr_{N (Γ)} (u · ∑_{δ′∈(δ)}(δ′)^{−1}) .

Theorem 2.12 Suppose that Γ is amenable. Then the following diagram commutes

K_{0}(ℂΓ)  ⟶^{HS}  class_{0}(Γ)
i↓  ^{r}↓
G_{0}(ℂΓ) ⟶^{j} K_{0}(N (Γ)) ⟶^{dim_{N (Γ)}^{u}} cent(N (Γ)) ⟶^{ϕ} class(Γ)_{cf}

where r is given by restriction and the other maps have been defined in 2.1, Lemma 2.3, 2.6, 2.10 and 2.11.

\underline{\underline{Proof}} : One has to show for an element A ∈ M(n, n, ℂΓ) and δ ∈ Γ such that (δ) is finite

tr_{ℂΓ}^{u}(A)(δ) = (ϕ ∘ tr_{N (Γ)}^{u}(A))(δ).  (2.13)

It suffices to show for γ ∈ Γ and δ ∈ Γ such that (δ) is finite

tr_{ℂΓ}^{u}(γ)(δ) = (ϕ ∘ tr_{N (Γ)}^{u}(γ))(δ).  (2.14)

The universal center-valued von Neumann trace satisfies for γ ∈ Γ

tr_{N (Γ)}^{u}(γ) = \{ \begin{matrix}|(γ)|^{−1} · ∑_{γ′∈(γ)} γ′ & if (γ) is finite \\ 0 & otherwise \end{matrix} .

This follows from the facts that tr_{N (Γ)}^{u} is ultraweakly continuous and the identity on the center of N (Γ) and that tr_{N (Γ)}^{u}(\frac{1}{n} · ∑_{i=1}^{n} δ_{i}) = tr_{N (Γ)}^{u}(δ) holds for elements δ_{1}, δ_{2}, …, δ_{n} in (δ). Notice that tr_{N (Γ)}(δ) is 1 if δ = 1 and 0 otherwise and that tr_{ℂΓ}^{u}(γ)(δ) = 0 if (δ) is finite and (γ) is infinite. Hence 2.14 and thus 2.13 follow from the computation for γ, δ ∈ Γ such that (γ) and (δ) are finite

ϕ(|(γ)|^{−1} · ∑_{γ′∈(γ)} γ′)(δ) = tr_{N (Γ)} ((|(γ)|^{−1} · ∑_{γ′∈(γ)} γ′) · ( ∑_{δ′∈(δ)}(δ′)^{−1}))
= ∑_{γ′∈(γ)} ∑_{δ′∈(δ)} |(γ)|^{−1} · tr_{N (Γ)}(γ′ · (δ′)^{−1})
= ∑_{γ′∈(γ),δ′∈(δ),γ′=δ′} |(γ)|^{−1}

= \{ \begin{matrix}1 & if (γ) = (δ) \\ 0 & otherwise \end{matrix}
= tr_{ℂΓ}^{u}(γ)(δ).

This finishes the proof of Theorem 2.12.

Lemma 2.15 Let Γ be a discrete group. Then there is a commutative diagram whose the left vertical arrow is an isomorphism

(colim_{Or(Γ,FIN)} K_{0}(ℂH)) ⊗_{ℤ} ℂ ⟶^{l} K_{0}(ℂΓ) ⊗_{ℤ} ℂ
h↓≅  ↓^{HS}
class_{0}(Γ)_{f}  ⟶_{e} class_{0}(Γ)

\underline{\underline{Proof}} : Firstly we explain the maps in the square. The colimit is taken for the covariant functor

Or(G, FIN) ⟶ ABEL G/H ↦ K_{0}(ℂH)

to the category of abelian groups which is given by induction. Here Or(G, FIN) is the full subcategory of the orbit category Or(G) consisting of objects G/H with finite H. The map l is induced by the universal property of the colimit and the various maps K_{0}(ℂH) ⟶ K_{0}(ℂΓ) induced by the inclusions of finite subgroups H of Γ in Γ. The map e is given by the inclusion con(Γ)_{f} ⟶ con(Γ).

Define for a group homomorphism ψ : Γ ⟶ Γ′ a map ψ_{∗} : con(Γ) ⟶ con(Γ′) by sending (h) to (ψ(h)). It induces a homomorphism ψ_{∗} : class_{0}(Γ) ⟶ class_{0}(Γ′). One easily checks that the following diagram commutes

K_{0}(ℂΓ) ⟶^{ψ_{∗}} K_{0}(ℂΓ′)
HS↓  ^{HS}↓
class_{0}(Γ) ⟶^{ψ_{∗}} class_{0}(Γ′)

(2.16)

There is a canonical isomorphism

f_{1} : (colim_{Or(Γ,FIN)} K_{0}(ℂH)) ⊗_{ℤ} ℂ ⟶^{≅} colim_{Or(Γ,FIN)} K_{0}(ℂH) ⊗_{ℤ} ℂ.  (2.17)

The Hattori-Stallings ranks for the various finite subgroups H of Γ induce an isomorphism

f_{2} : colim_{Or(Γ,FIN)} K_{0}(ℂH) ⊗_{ℤ} ℂ ⟶^{≅} colim_{Or(Γ,FIN)} class(H).  (2.18)

Let f′_{3} : colim_{Or(Γ,FIN)} con(H) ⟶ con(Γ)_{f} be the map induced by the the inclusions of the finite subgroups H of Γ. Define a map f′_{4} : con(Γ)_{f} ⟶ colim_{Or(Γ,FIN)} con(H) by sending (γ) ∈ con(Γ)_{f} to the image of (γ) ∈ con(⟨γ⟩) under the canonical structure map from con(⟨γ⟩) to colim_{Or(Γ,FIN)} con(H), where ⟨γ⟩ is the finite cyclic subgroup generated by γ. One easily checks that this is independent of the choice of the representative γ in (γ) and that f′_{3} and f′_{4} are inverse to one another. The bijection f′_{3} induces an isomorphism

f_{3} : colim_{Or(Γ,FIN)} class(H) ⟶ class(Γ)_{f},  (2.19)

because colimit and the functor sending a set to the complex vector space with this set as basis commute. Now the isomorphism h is defined as the composition of the isomorphisms f_{1} from 2.17, f_{2} from 2.18 and f_{3} from 2.19. It remains to check that the square in Lemma 2.15 commutes. This follows from the commutativity of 2.16. This finishes the proof of Lemma 2.15.

Corollary 2.20 Suppose that Γ is amenable. Then the image of the composition

G_{0}(ℂΓ) ⊗_{ℤ} ℂ ⟶^{j⊗_{ℤ}ℂ} K_{0}(N (Γ)) ⊗_{ℤ} ℂ ⟶^{dim_{N (Γ)}^{u}} cent(N (Γ)) ⟶^{ϕ} class(Γ)_{cf}

contains the complex vector space class_{0}(Γ)_{f,cf} with con(Γ)_{f} ∩ con(Γ)_{cf} as basis.

Remark 2.21 There is the conjecture that the canonical map

colim_{Or(Γ,FIN)} K_{0}(ℂH) ⟶^{≅} K_{0}(ℂΓ)

is bijective for all groups Γ. In particular this would imply by Lemma 2.15 that the Hattori-Stallings rank induces an isomorphism

HS : K_{0}(ℂΓ) ⊗_{ℤ} ℂ ⟶^{≅} class_{0}(Γ)_{f}.

Theorem 2.22 1. The map

l : (colim_{Or(Γ,FIN)} K_{0}(ℂH)) ⊗_{ℤ} ℂ ⟶ K_{0}(ℂΓ) ⊗_{ℤ} ℂ

is injective

2. If Γ is virtually polycyclic, then we obtain isomorphisms

HS : K_{0}(ℂΓ) ⊗_{ℤ} ℂ ⟶^{≅} class_{0}(Γ)_{f};
i : K_{0}(ℂΓ) ⟶^{≅} G_{0}(ℂΓ).

\underline{\underline{Proof}} : 1.) follows directly from Lemma 2.15.

2.) Moody has shown [23] that the obvious map ⊕_{H∈FIN} G_{0}(ℂH) ⟶ G_{0}(ℂΓ) given by induction is surjective. Since Γ is virtually polycyclic the complex group ring ℂΓ is regular, i.e. noetherian and any ℂΓ-module has a finite-dimensional projective resolution [24, Theorem 8.2.2 amd Theorem 8.2.20]. Now 1.) and Lemma 2.15 prove the claim.

Theorem 2.22.2 has already been proven in [5].

Remark 2.23 In particular we get from Theorem 2.12 that the map

ι : ℤ ⟶ G_{0}(ℂΓ) n ↦ [ℂΓ^{n}]

is injective, provided that Γ is amenable. It is likely that this property characterizes amenable groups. At least we can show for a group Γ which contains the free group F_{2} in two letters as subgroup, that ι is trivial by the following argument.

Induction with the inclusion F_{2} ⟶ Γ induces a homomorphism G_{0}(ℂF_{2}) ⟶ G_{0}(ℂΓ) which sends [ℂF_{2}] to [ℂΓ]. Hence it suffices to show [ℂF_{2}] = 0 in G_{0}(ℂF_{2}). The cellular chain complex of the universal covering of S^{1} ∨ S^{1} yields an exact sequence of ℂΓ-modules 0 → (ℂF_{2})^{2} → ℂF_{2} → ℂ → 0, where ℂ is equipped with the trivial F_{2}-action. Hence it suffices to show in [ℂ] = 0 in G_{0}(ℂF_{2}). Choose an epimorphism f : F_{2} ⟶ ℤ. Restriction with f defines a homomorphism G_{0}(ℂℤ) ⟶ G_{0}(ℂF_{2}). It sends ℂ viewed as trivial ℂℤmodule to ℂ viewed as trivial ℂF_{2}-module. Hence it remains to show [ℂ] = 0 in G_{0}(ℂℤ). This follows from the exact sequence 0 ⟶ ℂℤ ⟶^{s−1} ℂℤ ⟶ ℂ ⟶ 0 for s a generator of ℤ.

3. Groups with vanishing L^{2}-Betti numbers

In this section we investigate the following class of groups

Definition 3.1 Define the class of groups

B_{d} := {Γ | b_{p}^{(2)}(Γ) = 0 for 0 ≤ p ≤ d};
B_{∞} := {Γ | b_{p}^{(2)}(Γ) = 0 for 0 ≤ p}.

Notice that B_{0} is the class of infinite groups by [21, Theorem 4.10]. Definition 3.1 is motivated among other things by Corollary 4.5 and the following result.

Theorem 3.2 Let 1 ⟶ Δ ⟶ Γ ⟶ π ⟶ 1 be an exact sequence of groups. Suppose that Γ is finitely presented and one of the following conditions is satisfied.

1. |Δ| = ∞, b_{1}^{(2)}(Δ) < ∞ and π contains an element of infinite order or contains finite subgroups of arbitrary large order;

2. The ordinary first Betti number of Δ satisfies b_{1}(Δ) < ∞ and π belongs to B_{1}.

Then:

1. Let M be a closed oriented 4-manifold with Γ as fundamental group. Then

| sign(M)| ≤ χ(M);

2. Let def(Γ) be the deficiency, i.e. the maximum g(P) − r(P) for all presentations P where g(P) is the number of generators and r(P) the number of relations. Then

def(Γ) ≤ 1;

\underline{\underline{Proof}} : If the first condition is satisfied, then Γ belongs to B_{1} by Theorem 3.3.5. Now apply [18, Theorem 5.1 and Theorem 6.1 on page 212] .

Suppose that the second condition is satisfied. Let p : \overline{M} ⟶ M be the regular covering associated to Δ. There is a universal coefficient spectral sequence converging to H_{p+q}^{π}(\overline{M}; N (π)) with E_{p,q}^{2} = Tor_{p}^{ℂπ}(H_{q}(\overline{M}; ℂ), N (π)) [27, Theorem 5.6.4 on page 143]. Since H_{q}(\overline{M}; ℂ) is ℂ with the trivial π-action for q = 0 and finite-dimensional as complex vector space by assumption for q = 1, Theorem 1.4.4 and [21, Lemma 3.4.3] imply dim(E_{p,q}^{2}) = 0 for p + q = 1 and hence b_{1}^{(2)}(\overline{M}; N (π)) = 0. The arguments in [18, Theorem 5.1 and Theorem 6.1 on page 212] for the universal covering of M apply also to \overline{M}.

The idea to take another covering than the universal covering in the proof of Theorem 3.2 is taken from [10, Corollary 5.2 on page 391]. More information about results like Theorem 3.2 are given can be found in [11].

Theorem 3.3 Let d be a non-negative integer or d = ∞. Then:

1. The class B_{∞} contains all infinite amenable groups;

2. If Γ contains a normal subgroup Δ with Δ ∈ B_{d}, then Γ ∈ B_{d};

3. If Γ is the union of a directed system of subgroups {Γ_{i} | i ∈ I} such that each Γ_{i} belongs to B_{d}, then Γ ∈ B_{d};

4. Let 1 ⟶ Δ ⟶^{i} Γ ⟶^{p} π ⟶ 1 be an exact sequence of groups such that b_{p}^{(2)}(Δ) is finite for all p ≤ d. Suppose that Bπ has finite d-skeleton and that there is an injective endomorphism j : π ⟶ π whose image has finite index, but is not equal to π (for example π = ℤ^{n}). Then Γ ∈ B_{d};

5. Let 1 ⟶ Δ ⟶^{i} Γ ⟶^{p} π ⟶ 1 be an exact sequence of groups such that Δ ∈ B_{d−1}, b_{d}^{(2)}(Δ) < ∞ and π contains an element of infinite order or a finite subgroup of arbitrary large order. Then Γ ∈ B_{d};

6. Suppose that there are groups Γ_{1} and Γ_{2} and group homomorphisms ϕ_{i} : Γ_{0} ⟶ Γ_{i} for i = 1, 2 such that ϕ_{0} and ϕ_{1} are injective, Γ_{0} belongs to B_{d−1},Γ_{1} and Γ_{2} belong to B_{d} and Γ is the amalgamated product Γ_{1} ∗_{Γ_{0}} Γ_{2} with respect to ϕ_{1} and ϕ_{2}. Then Γ belongs to B_{d}.

\underline{\underline{Proof}} : 1.) follows from 0.7 or [Theorem 0.2 on page 191]Cheeger-Gromov (1986).

2.) We obtain a fibration BΔ ⟶ BΓ ⟶ Bπ for π = Γ/Δ. There is the Leray-Serre spectral sequence converging to H_{p+q}^{Γ}(EΓ, N (Γ)) with E_{p,q}^{1} = H_{q}^{Δ}(EΔ; N (Γ)) ⊗_{ℤπ} C_{p}(Eπ) for an appropriate ℤπ-action on H_{q}^{Δ}(EΔ; N (Γ)) coming from the fiber transport. Because of Additivity (see Theorem 1.4.4) it suffices to show for p + q ≤ d

dim_{N (Γ)} (E_{p,q}^{1}) = 0.  (3.4)

Since b_{q}^{(2)}(Δ) = dim_{N (Γ)} (H_{q}^{Δ}(EΔ; N (Γ))) by [21, Theorem 4.9] and C_{p}(Eπ) is a direct sum of copies of ℤπ, Cofinality (see Theorem 1.4.4) proves 3.4.

3.) Using for instance the bar-resolution model for EΓ, one gets that EΓ is the colimit of a directed system of subspaces of the form EΓ_{i} ×_{Γ_{i}} Γ directed by I. Hence

H_{p}^{Γ}(EΓ; N (Γ)) = colim_{i∈I} H_{p}^{Γ}(EΓ_{i} ×_{Γ_{i}} Γ; N (Γ)).

Since dim_{N (Γ)} (H_{p}^{Γ}(EΓ_{i} ×_{Γ_{i}} Γ; N (Γ))) = b_{p}^{(2)}(Γ_{i}) by [21, Theorem 4.9] the claim follows from [21, Theorem 2.9].

4.) Fix an integer n ≥ 1. Put Γ′ = p^{−1}(im(j^{n})). If k is the index of im(j) in π, then k^{n} is the index of im(j^{n}) in π and of Γ′ in Γ and we conclude

b_{p}^{(2)}(Γ) = \frac{b_{p}^{(2)}(Γ′)}{k^{n}}.  (3.5)

Since im(j^{n}) is isomorphic to π, we have an exact sequence 1 ⟶ Δ ⟶ Γ′ ⟶ π ⟶ 1. Let i_{p} be the number of p-cells in Bπ. We get from the Leray-Serre spectral sequence and Additivity (see Theorem 1.4)

b_{p}^{(2)}(Γ′) ≤ ∑_{n=0}^{p} b_{q}^{(2)}(Δ) · i_{p−q}.  (3.6)

Equations 3.5 and 3.6 imply

b_{p}^{(2)}(Γ) = \frac{∑_{q=0}^{p} b_{q}^{(2)}(Δ) · i_{p−q}}{k^{n}} .  (3.7)

Since k > 1 and 3.7 holds for all n ≥ 1 and ∑_{q=0}^{p} b_{q}^{(2)}(Δ) · i_{p−q} is finite for p ≤ d by assumption, the claim follows.

5.) Using the spectral sequence which converges to H_{p+q}^{Γ}(EΓ; N (Γ)) and has as E^{2}-term E_{p,q}^{2} = H_{p}^{π}(Eπ; H_{q}^{Δ}(EΔ; N (Γ))) the proof of assertion 5.) is reduced to the proof of

dim_{N (Γ)} (H_{0}^{π}(Eπ; H_{d}^{Δ}(EΔ; N (Γ)))) = 0,  (3.8)

since dim_{N (Γ)} (H_{q}^{Δ}(EΔ; N (Γ))) = b_{q}^{(2)}(Δ) by [21, Theorem 4.9] and hence vanishes for q < d by assumption. Let π′ ⊂ π be a subgroup (not necessarily normal). Let Γ′ ⊂ Γ be the preimage of π′ under the canonical projection Γ ⟶ π. Then we obtain an exact sequence 1 ⟶ Δ ⟶ Γ′ ⟶ π′ ⟶ 1. We have

H_{0}^{π′}(Eπ′; H_{d}^{Δ}(EΔ; N (Γ′))) = H_{d}^{Δ}(EΔ; N (Γ′)) ⊗_{ℂ[π′]} ℂ;
H_{0}^{π}(Eπ; H_{d}^{Δ}(EΔ; N (Γ))) = H_{d}^{Δ}(EΔ; N (Γ)) ⊗_{ℂ[π]} ℂ;

Since H_{d}^{Δ}(EΔ; N (Γ)) ⊗_{ℂ[π]} ℂ is a quotient of H_{d}^{Δ}(EΔ; N (Γ)) ⊗_{ℂ[π′]} ℂ we conclude from Additivity (see Theorem 1.4) and from [21, Theorem 4.9]

dim_{N (Γ)} (H_{d}^{Δ}(EΔ; N (Γ)) ⊗_{ℂ[π]} ℂ) ≤ dim_{N (Γ)} (H_{d}^{Δ}(EΔ; N (Γ)) ⊗_{ℂ[π′]} ℂ);
dim_{N (Γ′)} (H_{d}^{Δ}(EΔ; N (Γ′)) ⊗_{ℂ[π′]} ℂ) = dim_{N (Γ)} (H_{d}^{Δ}(EΔ; N (Γ)) ⊗_{ℂ[π′]} ℂ) .

This implies

dim_{N (Γ)} (H_{0}^{π}(Eπ; H_{d}^{Δ}(EΔ; N (Γ)))) ≤ dim_{N (Γ′)} (H_{0}^{π′}(Eπ′; H_{d}^{Δ}(EΔ; N (Γ′)))) .

Hence 3.8 would follow if we can find for each ϵ < 0 a subgroup π′ ⊂ π satisfying

dim_{N (Γ′)} (H_{0}^{π′}(Eπ′; H_{d}^{Δ}(EΔ; N (Γ′)))) ≤ ϵ.  (3.9)

We begin with the case where π′ is ℤ. From assertion 4 we conclude

dim_{N (Γ′)} (H_{p}^{Γ′}(EΓ′; N (Γ′))) = 0  for p ≤ d.  (3.10)

The Leray-Serre spectral sequence associated to 1 ⟶ Δ ⟶ Γ′ ⟶ ℤ ⟶ 1 has an E^{2}- term which satisfies E_{p,q}^{2} = 0 for q ≠ 0, 1 since Bℤ has the 1-dimensional model S^{1}. Since it converges to H_{p+q}^{Γ′}(EΓ′; N (Γ′)), we conclude 3.9 for ϵ = 0 from 3.10 and Additivity (see Theorem 1.4). Now suppose π′ is finite. Then we get

dim_{N (Γ′)} (H_{0}^{π′}(Eπ′; H_{d}^{Δ}(EΔ; N (Γ′)))) = dim_{N (Γ′)} (H_{d}^{Γ′}(EΓ′; N (Γ′)))
= b_{d}^{(2)}(Γ′)
= \frac{b_{d}^{(2)}(Δ)}{|π′|}.

If we can find π′ with arbitrary large |π′| we get 3.9.

6.) One easily checks using the Seifert-van Kampen Theorem, that there is a Γ-push out

Γ ×_{Γ_{0}} EΓ_{0} ⟶ Γ ×_{Γ_{1}} EΓ_{1}
↓  ↓
Γ ×_{Γ_{2}} EΓ_{2} ⟶  EΓ

We conclude from [21, Theorem 4.9] dim (H_{p}^{Γ}(EΓ_{i} ×_{Γ_{i}} Γ; N (Γ))) = b_{p}^{(2)}(Γ_{i}). Now the claim follows from Additivity (see Theorem 1.4.4) and the a long exact homology sequence for H_{∗}^{Γ}(−, N (Γ)). This finishes the proof of Theorem 3.3.

So far we have no example with negative answer to the following question and can give an affirmative answer in some special cases.

Question 3.11 Let 1 ⟶ Δ ⟶ Γ ⟶ π ⟶ 1 be an exact sequence such that b_{p}^{(2)}(Δ) < ∞ for all p ≥ 0 and π belongs to B_{∞}. Does then Γ belong to B_{∞}? Using Theorem 3.3 one can prove this for instance if π is elementary amenable.

More generally, if F ⟶ E ⟶ B is a fibration such that b_{p}^{(2)}(F̃; N (π_{1}(E))) < ∞ and b_{p}^{(2)}(B̃; N (π_{1}(B))) = 0 holds for p ≥ 0, does then b_{p}^{(2)}(Ẽ; N (π_{1}(E))) = 0 hold for p ≥ 0?

Remark 3.12 Compact 3-manifolds whose fundamental groups belong to B_{∞} are characterized in [16, Proposition 6.5 on page 54]. The generalized Singer-Conjecture says that for an aspherical closed manifold M all the L^{2}-Betti numbers of its universal covering vanish possibly except in the middle dimension. In particular it implies that the fundamental group of an aspherical closed odd-dimensional manifold belongs to B_{∞}. Thompson’s group F belongs to B_{∞} [19, Theorem 0.8]. More information about the class B_{1} is given in [3].

4. L^{2}-Euler characteristics and the Burnside group

In this section we extend some of the results [6] about L^{2}-Euler characteristics and investigate the Burnside group of a discrete group Γ. This extends the classical notions of the Burnside ring, Burnside ring congruences and equivariant Euler characteristics for finite groups.

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

Definition 4.1 Let X be a (left) Γ-space and V be a A-ℤΓ-module. Define

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

The condition h(X; V) < ∞ ensures that the sum defining χ^{(2)}(X; V) converges and that χ^{(2)}(X; V) satisfies the usual additivity formula, i.e. for a Γ-CW -complex X with Γ-CW -subcomplexes X_{0}, X_{1} and X_{2} satisfying X = X_{1} ∪ X_{2}, X_{0} = X_{1} ∩ X_{2} and h(X_{k}; V) < ∞ for k = 0, 1, 2 one has

h(X; V) < ∞;  (4.2)
χ^{(2)}(X; V) = χ^{(2)}(X_{1}; V) + χ^{(2)}(X_{2}; V) − χ^{(2)}(X_{0}; V).  (4.3)

The next theorem generalizes [6, Theorem 0.3 on page 191]

Theorem 4.4 Let X and Y be Γ-CW -complexes such that m(X) < ∞ and m(Y) < ∞ holds. Then

1.

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

2. Suppose that Γ is amenable. Then

∑_{c∈I(X)}(−1)^{dim(c)} · |Γ_{c}|^{−1} = ∑_{p≥0}(−1)^{p} · dim (N (Γ) ⊗_{ℂΓ} H_{p}(X; ℂ)) ,

where H_{p}(X; ℂ) is the cellular or the singular homology of X with complex coefficients. In particular ∑_{c∈I(X)}(−1)^{dim(c)} · |Γ_{c}|^{−1} depends only the ℂΓ-isomorphism class of the ℂΓ-modules H_{n}(X; ℂ) for all n ≥ 0; 3. If for all c ∈ I(X) the group Γ_{c} is finite or belongs to the class B_{∞}, then

b_{p}^{(2)}(X; N (Γ)) = b_{p}^{(2)}(EΓ × X; N (Γ))  for p ≥ 0;
χ^{(2)}(X; N (Γ)) = χ^{(2)}(EΓ × X; N (Γ));
∑_{c∈I(X)}(−1)^{dim(c)} · |Γ_{c}|^{−1} = χ^{(2)}(EΓ × X; N (Γ)).

4. Suppose that f : X ⟶ Y is a Γ-equivariant map, such that the induced map H_{p}(f; ℂ) on the singular or cellular homology with complex coefficients is bijective. Suppose that for all c ∈ I(X) and c ∈ I(Y) the group Γ_{c} is finite or belongs to the class B_{∞}. Then

∑_{c∈I(X)}(−1)^{dim(c)} · |Γ_{c}|^{−1} = ∑_{c∈I(Y)}(−1)^{dim(c)} · |Γ_{c}|^{−1}.

\underline{\underline{Proof}} : 1.) Additivity and Cofinality (see Theorem 1.4) and [21, Lemma 3.4.1] imply

dim(C_{p}(X; N (Γ))) = ∑_{c∈I(X),dim(c)=p} |Γ_{c}|^{−1};
dim(H_{p}^{Γ}(X; N (Γ))) ≤ dim(C_{p}(X; N (Γ)));
∑_{p≥0}(−1)^{p} · dim(C_{p}(X; N (Γ))) = χ^{(2)}(X; N (Γ))).

2.) follows from the first assertion and 0.6.

3.) Because of the first equation it suffices to prove that the dimension of the kernel and the cokernel of the map induced by the projection

pr_{∗} : H_{p}^{Γ}(EΓ × X; N (Γ)) ⟶ H_{p}^{Γ}(X; N (Γ))

are trivial. Notice that X is the colimit of its finite Γ-subcomplexes. Since H_{p}^{Γ}(−, N (Γ)) is compatible with colimits and colimit preserves exact sequences, we can assume by Theorem [21, Theorem 2.9] and Additivity (see Theorem 1.4) that X itself is finite. By induction over the number of equivariant cells, the long exact homology sequence and Additivity (see Theorem 1.4) the claim reduces to the case where X is of the shape Γ/H. Because of [21, Theorem 4.9] it suffices to prove for the map pr_{∗} : H_{p}^{H}(EH; N (H)) ⟶ H_{p}^{H}({∗}; N (H)) that its kernel and cokernel have trivial dimension, provided that H is finite or belongs to B_{∞}. This is obvious for finite H and follows for H ∈ B_{∞} from the definition of B_{∞} and [21, Theorem 4.10].

4.) Since EΓ×X is free and the map id ×f : EΓ × X ⟶ EΓ × Y induces an isomorphism on singular homology it induces an isomorphism H_{p}^{Γ}(EΓ × X; N (Γ)) ⟶ H_{p}^{Γ}(EΓ × Y; N (Γ)) and we conclude χ^{(2)}(EΓ × X; N (Γ)) = χ^{(2)}(EΓ × Y; N (Γ)). Now assertion 4.) follows from assertion 3.). This finishes the proof of Theorem 4.4.

As explained in [6, Proposition 0.4 on page 192] the L^{2}-Euler characteristic extends the notion of the virtual Euler characteristic which is due to Wall [25]. Information about this notion can be found for instance in [4, chaper IX]. The next result generalizes [6, Corollary 0.6 on page 193] Corollary 4.5 Let Γ be a group belonging to B_{∞}. Then χ^{2}(EΓ; N (Γ)) is defined and vanishes. If its virtual Euler characteristic χ_{virt}(Γ) is defined, then it vanishes. In particular χ(BΓ) vanishes if BΓ can be choosen to be a finite CW -complex.

Next we introduce the Burnside group and the equivariant Euler characteristic. The elementary proof the following lemma is left to the reader.

Lemma 4.6 Let H and K be subgroups of Γ. Let NK be the normalizer of K in Γ and WK be NK/K. Then

1. Γ/H^{K} = {gH | g^{−1}Kg ⊂ H};

2. The map

ϕ : Γ/H^{K} ⟶ consub(H) gH ↦ g^{−1}Kg

induces an injection

WK\(Γ/H^{K}) ⟶ consub(H),

where consub(H) is the set of conjugacy classes in H of subgroups of H;

3. The WK-isotropy group of gH ∈ Γ/H^{K} is (gHg^{−1} ∩ NK)/K ⊂ NK/K = WK;

4. If H is finite, then Γ/H^{K} is a finite union of WK-orbits of the shape WK/L for finite subgroups L ⊂ WK.

Definition 4.7 Define the Burnside group A(Γ) by the Grothendieck group of the abelian monoid under disjoint union of Γ-isomorphism classes of proper cocompact Γ-sets S, i.e. Γ-sets S for which the isotropy group of each element in S and the quotient Γ\S are finite.

Notice that A(Γ) is the free abelian group generated by Γ-isomorphism classes of orbits Γ/H for finite subgroups H ⊂ Γ and that Γ/H and Γ/K are Γ-isomorphic if and only if H and K are conjugate in Γ. If Γ is a finite group, A(Γ) is the classical Burnside ring [8, section 5], [9, chapter IV]. If Γ is infinite, then the cartesian product of two proper cocompact Γ-sets with the diagonal action is not cocompact any more so that the cartesian product does not induce a ring structure on A(Γ). At least there is a bilinear map induced by the cartesian product A(Γ_{1}) ⊗ A(Γ_{2}) ⟶ A(Γ_{1} × Γ_{2}).

Definition 4.8 Let X be a proper finite Γ-CW -complex. Define its equivariant Euler characteristic

χ^{Γ}(X) := ∑_{c∈I(X)}(−1)^{dim(c)} · [Γ/Γ_{c}] ∈ A(Γ).

An additive invariant (A, a) for proper finite Γ-CW -complexes X consists of an abelian group A and a function a which assigns to any proper finite Γ-CW -complex X an element a(X) ∈ A such that the following three conditions hold, i.) if X and Y are Γ-homotopy equivalent, then a(X) = a(Y), ii.) if X_{0},X_{1} and X_{2} are Γ-CW -subcomplexes of X with X = X_{1} ∪ X_{2} and X_{0} = X_{1} ∩ X_{2}, then a(X) = a(X_{1}) + a(X_{2}) − a(X_{0}), and iii.) a(∅) = 0. We call an additive invariant (U, u) universal, if for any additive invariant (A, a) there is precisely one homomorphism ψ : U ⟶ A such that ψ(u(X)) = a(X)) holds for all proper finite Γ-CW -complexes. One easily checks using induction over the number of equivariant cells

Lemma 4.9 (A(Γ), χ^{Γ}) is the universal additive invariant for finite proper Γ-CW -complexes.

Definition 4.10 Define for a finite subgroup K ⊂ Γ the L^{2}-character map

ch_{K}^{Γ} : A(Γ) ⟶ ℚ [S] ↦ ∑_{i=1}^{r} |L_{i}|^{−1}

if WK/L_{1}, WK/L_{2}, … WK/L_{r} are the WK-orbits of S^{K}. Define the global L^{2}-character map by

ch^{Γ} := ∏_{(K)} ch_{K}^{Γ} : A(Γ) ⟶ ∏_{(K)} ℚ

where (K) runs over the conjugacy classes of finite subgroups of Γ.

Lemma 4.11 Let X be a finite proper Γ-CW -complex and K ⊂ Γ be a finite subgroup. Then X^{K} is a finite proper WK-CW -complex and

χ^{(2)}(X^{K}; N (WK)) = ch_{K}^{Γ}(χ^{Γ}(X)).

\underline{\underline{Proof}} : The WK-space X^{K} is a finite proper WK-CW -complex because for finite H ⊂ Γ the WK-set Γ/H^{K} is proper and cocompact by Lemma 4.6. Since the assignment which associates to a finite proper Γ-CW -complex X the element χ^{(2)}(X^{K}; N (WK)) in ℚ is an additive invariant, it suffices by Lemma 4.9 to check the claim for X = Γ/H for finite H ⊂ Γ. Then the claim follows from the fact that χ^{(2)}(WK/L; N (WK)) = |L|^{−1} holds for finite L ⊂ WK.

Notice that one gets from Lemma 4.6 the following explicit formula for the value of ch_{K}^{Γ}(Γ/H). Namely, define

L_{K}(H) := {(L) ∈ consub(H) | L conjugated to K in Γ}.

For (L) ∈ L_{K}(H) choose L ∈ (L) and g ∈ Γ with g^{−1}Kg = L. Then

g(H ∩ NL)g^{−1} = gHg^{−1} ∩ NK;
|(gHg^{−1} ∩ NK)/K|^{−1} = \frac{|K|}{|H ∩ NL|}.

This implies

ch_{K}^{Γ}(Γ/H) = ∑_{(L)∈L_{K}(H)} \frac{|K|}{|H ∩ NL|}.  (4.12)

Lemma 4.13 The global L^{2}-character map of Definition 4.10 induces a map denoted by

ch^{Γ} ⊗_{ℤ}ℚ : A(Γ) ⊗_{ℤ} ℚ ⟶ ∏_{(K)} ℚ.

It is injective. If Γ has only finitely many conjugacy classes of finite subgroups, then it is bijective.

\underline{\underline{Proof}} : Consider an element ∑_{i=1}^{n} r_{i} · [Γ/H_{i}] in the kernel of ch^{Γ} ⊗_{ℤ}ℚ. We show by induction over n that the element must be trivial. The begin n = 0 is trivial, the induction step done as follows.We can choose the numeration such that H_{i} subconjugated to H_{j} implies i ≥ j. We get from 4.12

ch_{K}^{Γ}(Γ/H) = 1  , if H = K;
ch_{K}^{Γ}(Γ/H) = 0  , if K is not subconjugated to H in Γ.

This implies

ch_{H_{1}}^{Γ} (∑_{i=1}^{n} r_{i} · [Γ/H_{i}]) = r_{1}

and hence r_{1} = 0. Hence the global L^{2}-character map is injective. If Γ has only finitely many conjugacy classes of finite subgroups, then the source and target of ch^{Γ} ⊗_{ℤ}ℚ are rational vector spaces of the same finite dimension and ch^{Γ} ⊗_{ℤ}ℚ must be bijective.

Remark 4.14 Suppose that there are only finitely many conjugacy classes (H_{1}), (H_{2}), …, (H_{r}) of finite subgroups in Γ. Without loss of generality we can assume that H_{i} subconjugated to H_{j} implies i ≥ j. With respect to the obvious ordered basis for the source and target the map ch^{Γ} ⊗_{ℤ}ℚ is described by an upper triangular matrix A with ones on the diagonal. One can get an explicit inverse A^{−1} which again has ones on the diagonal. This leads to a characterization of the image of A(Γ) under the global L^{2}-character map χ^{Γ}. Namely, an element in η ∈ ∏_{i=1}^{r} ℚ lies in ch^{Γ}(A(Γ)) if and only if the following Burnside integrality conditions are satisfied

A^{−1}η ∈ ∏_{i=1}^{r} ℤ.  (4.15)

Now suppose that Γ is finite. Then the global L^{2}-character map is related to the classical character map by the factor |WK|^{−1}, i.e. we have for each subgroup K of Γ and any finite Γ-set S

ch_{K}^{Γ}(S) = |WK|^{−1} · |S^{K}|.  (4.16)

One easily checks that under the identification 4.16 the integrality conditions 4.15 correspond to the classical Burnside ring congruences for finite groups [8, section 5.8], [9, section IV.5].

Let E(Γ, FIN) be the classifying Γ-space for the family FIN of finite subgroups. This Γ-CW -complex is characterized up to Γ-homotopy by the property that its H-fixed point set is contractible if H ⊂ Γ is finite and empty otherwise. It is also called the classifying space for proper Γ-spaces and denoted by \underline{E}Γ in the literature. For more information about E(Γ, FIN) we refer for instance to [2], [7, section 7], [9, section I.6],

Lemma 4.17 Suppose that there is model for E(Γ, FIN) which is a finite Γ-CW -complex. Then there are only finitely many conjugacy classes of finite subgroups and for a finite subgroup K ⊂ Γ

ch_{K}^{Γ}(χ^{Γ}(E(Γ, FIN))) = χ^{(2)}(WK).

If Γ is amenable, then we get for a finite subgroup K ⊂ Γ

ch_{K}^{Γ}(χ^{Γ}(E(Γ, FIN))) = |WK|^{−1},

where |WK|^{−1} is to be understood as 0 for infinite WK.

\underline{\underline{Proof}} : We get from Lemma 4.11 since E(Γ, FIN)^{K} is a model for E(WK, FIN)

ch_{K}^{Γ}(χ^{Γ}(E(Γ, FIN))) = χ^{(2)}(E(WK, FIN); N (WK)).

Now apply Theorem 4.4.3. and 0.7.

Example 4.18 Let 1 ⟶ ℤ^{n} ⟶ Γ ⟶ ℤ/p ⟶ 1 be an extension of groups for n ≥ 1 and a prime number p. Then E(Γ, FIN) can be choosen as a finite Γ-CW -complex because only the following cases can occur. If Γ contains a finite subgroup, then Γ is a semi-direct product of ℤ^{n} and ℤ/p and one can construct a finite Γ-CW -complex as model for E(Γ, FIN) with ℝ^{n} as underlying space. If the group Γ contains no finite subgroup, one shows inductively over n that there is a finite model for BΓ. In the induction step use the fact that Γ can be written as an extension of a group for which the induction hypothesis applies and ℤ. We want to compute χ^{Γ}(E(Γ, FIN)). The conjugation action of Γ on the normal subgroup ℤ^{n} factorizes through the projection Γ ⟶ ℤ/p into a operation ρ of ℤ/p onto ℤ^{n}. If this operation has a non-trivial fixed point, then WH is infinite for any finite subgroup H of Γ and we conclude from Lemma 4.13 and Theorem 4.17 that

χ^{Γ}(E(Γ, FIN)) = 0.

Now suppose that this operation ρ has no non-trivial fixed points. Let H_{0} be the trivial subgroup and H_{1}, H_{2}, …, H_{r} be a complete set of representatives of the conjugacy classes of finite subgroups. Each H_{i} is isomorphic to ℤ/p. One easily checks that there is a bijection

H^{1}(ℤ/p; ℤ_{ρ}^{n}) ⟶ {(H) | H ⊂ Γ, 1 < |H| < ∞}

and in particular r ≥ 1, where ℤ_{ρ}^{n} denotes the ℤ[ℤ/p]-module given by ℤ^{n} and ρ. We compute using 4.12

ch_{H_{0}}^{Γ}(Γ/H_{0}) = 1;
ch_{H_{0}}^{Γ}(Γ/H_{j}) = \frac{1}{p}  j = 1, 2, … r;
ch_{H_{i}}^{Γ}(Γ/H_{j}) = 1  i = j, i, j = 1, 2, … r;
ch_{H_{i}}^{Γ}(Γ/H_{j}) = 0  i ≠ j, i, j = 1, 2, … r; .

We conclude

χ^{Γ}(E(Γ, FIN)) = −\frac{r}{p} · [Γ/H_{0}] + ∑_{i=1}^{r} [Γ/H_{i}].

The integrality conditions of 4.15 become in this case

η_{0} − \frac{1}{p} · ∑_{i=1}^{r} η_{i} ∈ ℤ;
η_{i} ∈ ℤ  i = 1, 2 , … r.

5. Values of L^{2}-Betti numbers

In this section we investigate the possible values of L^{2}-Betti numbers.

Conjecture 5.1 Let Γ be a group and let X be a free finite Γ-CW -complex. Then

b_{p}^{(2)}(X; N (Γ)) ∈ ℚ.

If d is a positive integer such that the order of any finite subgroup of Γ divides d, then

d · b_{p}^{(2)}(X; N (Γ)) ∈ ℤ.

The significance of Conjecture 5.1 and its relation to a question of Atiyah [1, page 72] about the rationality of the analytic L^{2}-Betti numbers of 0.4 are explained in [20, Section 2]. Let C be the smallest class of groups which contains all free groups, is closed under directed unions and satisfies G ∈ C whenever G contains a normal subgroup H such that H belongs to C and G/H is elementary amenable. Conjecture 5.1 has been proven for groups Γ ∈ C by Linnell [14], provided that there is an upper bound on the order of finite subgroups of Γ.

Theorem 5.2 1. Let Γ be a group such that there is no bound on the order of finite subgroups. Then

(a) Given β ∈ [0, ∞], there is a countably generated projective ℤΓ-module P satisfying

dim_{N (Γ)}(N (Γ) ⊗_{ℤΓ} P) = β;

(b) Given a sequence β_{3}, β_{4}, … of elements in [0, ∞], there is a free Γ-CW -complex

X satisfying

b_{p}^{(2)}(X; N (Γ)) = β_{p} for p ≥ 3;

If Γ is countably presented, one can arrange that X has countably many Γequivariant cells; 2. Let Γ be a group such that there is a bound on the order of finite subgroups. Let d be the

least common multiple of the orders of finite subgroups of Γ. Suppose that Conjecture 5.1 holds for Γ. Then we get for any Γ-space X and p ≥ 0

d · b_{p}^{(2)}(X; N (Γ)) ∈ ℤ ∪ {∞};

3. Given a sequence of elements β_{1}, β_{2}, … [0, ∞], there is a countable group Γ with b_{p}^{(2)}(Γ) = β_{p} for p ≥ 1. If β_{1} is rational, Γ can be chosen to be finitely generated.

\underline{\underline{Proof}} : 1.a.) Since there is no bound on the order of finite subgroups, we can find a sequence of finite subgroups H_{1}, H_{2}, … of Γ such that β = ∑_{i=1}^{∞} |H_{i}|^{−1}. Then

P = ⊕_{i=1}^{∞}ℂ[Γ/H_{i}]

is the desired module by [21, Lemma 3.4.1] and Additivity and Cofinality (see Theorem 1.4.4.).

1.b.) By assertion 1.a) we can choose a sequence of countably generated projective ℂΓmodules P_{3}, P_{4}, … such that for p ≥ 3

dim_{N (Γ)}(N (Γ) ⊗_{ℤΓ} P_{p}) = β_{p}.  (5.3)

Next we construct inductively a nested sequence X_{2} ⊂ X_{3} ⊂ … of Γ-CW -complexes together with Γ-retractions r_{p} : X_{p} ⟶ X_{p−1} for p ≥ 3 such that X_{2} is the 2-skeleton of a model for EΓ, X_{p} is obtained from X_{p−1} by attaching countably many free Γ-equivariant p-cells and p + 1-cells and

H_{n}(X_{p}, X_{p−1}) = \{ \begin{matrix}P_{p} & n = p \\ 0 & n ≠ p \end{matrix} .  (5.4)

The Eilenberg-swindle yields a split-exact sequence 0 ⟶ C_{p+1} ⟶^{c_{p+1}} C_{p} ⟶ P_{p} ⟶ 0 of ℂΓmodules such that C_{p+1} and C_{p} are countably generated free ℤΓ-modules with a basis. Now one attaches for each element of the basis of C_{p} trivially a free Γ-equivariant p-cell to X. Then one attaches for each element of the basis of C_{p+1} a free Γ-equivariant p + 1-cell to X, where the attaching maps are choosen such that the cellular ℂΓ-chain complex of (Y, X) is just the ℤΓ-chain complex which is concentrated in dimension p + 1 and p and given there by C_{p+1} ⟶^{c_{p+1}} C_{p}. Details of the construction of the Γ-CW -complexes X_{p} and Γ-retractions r_{p} can be found in [22, Theorem 2.2, page 201], [26]. Now define Y = colim_{p→∞} X_{p}. One easily checks for p ≥ 3

H_{p}^{Γ}(Y; N (Γ)) = N (Γ) ⊗_{ℤΓ} P_{p};
b_{p}^{(2)}(Y; N (Γ)) = β_{p}.

2.) Let f : Y ⟶ X be a Γ-CW -approximation of X [17, page 35], i.e. a Γ-CW -complex Y with a Γ-map f such that f^{H} is a weak homotopy equivalence and hence a weak homology equivalence [28, Theorem IV.7.15 on page 182] for H ⊂ Γ. We get from [21, Lemma 4.8.2] that b_{p}^{(2)}(Y; N (Γ)) = b_{p}^{(2)}(X; N (Γ)) holds for all p ≥ 0. Hence we can assume without loss of generality that X is a Γ-CW -complex.

Conjecture 5.1 is equivalent to the statement that for any finitely presented ℤΓ-module M

d · dim_{N (Γ)}(N (Γ) ⊗_{ℤΓ} M) ∈ ℤ.  (5.5)

This follows essentially from [20, Lemma 2.2]. Now let D_{∗} be any ℤΓ-chain complex such that D_{p} is isomorphic to ⊕_{i=1}^{r}ℤ[Γ/H_{i}] for some non-negative integer r and finite subgroups H_{i}. Then the cokernel of each of the differentials d_{p} is a finitely presented ℤΓ-module and 5.5 yields for all p ≥ 0 using Additivity (see Theorem 1.4.4)

d · dim_{N Γ)} (cok (id_{N (Γ)} ⊗_{ℤΓ}d_{p})) ∈ ℤ;
d · dim_{N Γ)} (im (id_{N (Γ)} ⊗_{ℤΓ}d_{p})) ∈ ℤ;
d · dim_{N Γ)} (ker (id_{N (Γ)} ⊗_{ℤΓ}d_{p})) ∈ ℤ;
d · dim_{N Γ)} (H_{p}(N (Γ) ⊗_{ℤΓ} D_{∗})) ∈ ℤ.  (5.6)

Let X^{∞} be the Γ-CW -subcomplex of X consisting of points whose isotropy groups are infinite. The sequence 0 ⟶ C_{∗}(X^{∞}) ⟶ C_{∗}(X) ⟶ C_{∗}(X, X^{∞}) ⟶ 0 of cellular ℤΓ-chain complexes is exact. Since it is ℤΓ-split exact in each dimension, the sequence obtained by tensoring with N (Γ) is still exact. The associated long exact homology sequence, Additivity (see Theorem 1.4.4) [21, Lemma 3.4.1] imply for p ≥ 0

dim_{N (Γ)}(N (Γ) ⊗_{ℤΓ} C_{p}(X^{∞})) = 0;  (5.7)
b_{p}^{(2)}(X; N (Γ)) = dim_{N (Γ)} (H_{p}(N (Γ) ⊗_{ℤΓ} C_{∗}(X, X^{∞}))) .  (5.8)

Notice that C_{p}(X, X^{∞}) is a sum of ℤΓ-modules of the shape ℤ[Γ/H] for finite groups H ⊂ Γ. Hence C_{∗}(X, X^{∞}) is a colimit over a directed set I of subcomplexes D_{∗}[i] (directed by inclusion) such that each D_{p}[i] is isomorphic to ⊕_{i=1}^{r}ℤ[Γ/H_{i}] for some non-negative integer r and finite subgroups H_{i}. Since homology commutes with colimits we conclude from 5.8 and [21, Theorem 2.9.2]

b_{p}^{(2)}(X; N (Γ)) = sup{inf{dim_{N (Γ)}(im(H_{p}(N (Γ) ⊗_{ℤΓ} D_{∗}[i]) ⟶ H_{p}(N (Γ) ⊗_{ℤΓ} D_{∗}[j])))
| j ∈ I, i ≤ j} | i ∈ I}.

Since the set {r ∈ ℝ | d · r ∈ ℤ} is discrete in ℝ, it suffices to show for each inclusion ι : D_{∗}[i] ⟶ D_{∗}[j] and all p ≥ 0

d · dim_{N (Γ)} (im(ι_{∗} : H_{p}(N (Γ) ⊗_{ℤΓ} D_{∗}[i]) ⟶ H_{p}(N (Γ) ⊗_{ℤΓ} D_{∗}[j]))) ∈ ℤ.  (5.9)

Let F_{∗} be any acyclic N (Γ)-chain complex with F_{p} = 0 for p < 0 such that d · dim(F_{p}) ∈ ℤ holds for all p ≥ 0. Then we get d · dim(im(f_{p} : F_{p} ⟶ F_{p−1})) ∈ ℤ for all p ≥ 0 since we have the short exact sequences 0 ⟶ im(f_{p+1}) ⟶ F_{p} ⟶ im(f_{p}) ⟶ 0 and im(f_{1}) = F_{0}. Hence we obtain 5.9 from 5.6 and the conclusion above for the case where F is the long exact homology sequence of the pair (cyl(ι), D_{∗}[i]) since there is a ℤΓ-chain homotopy equivalence from the mapping cylinder cyl(ι) to D_{∗}[j] whose composition with the inclusion of D_{∗}[i] in cyl(ι) is ι.

3.) is proven in [6, section 4]. This finishes the proof of Theorem 5.2.

Remark 5.10 The group Γ = ∏_{i=1}^{∞} ℤ ∗ ℤ satisfies H_{p}^{Γ}(EΓ; N (Γ)) = 0 for all p ≥ 0. This is interesting in connection with the zero-in-the-spectrum conjecture ([15], [20, section 11]).

References

[1] Atiyah, M.: “Elliptic operators, discrete groups and von Neumann algebras”, Asterisque 32, 43 - 72 (1976)

[2] Baum, P., Connes, A., and Higson, N.: “Classifying space for proper actions and K-theory of group C^{∗}-algebras”, in: Doran, R.S. (ed.), “C^{∗}-algebras”, Contemporary Mathematics 167, 241–291 (1994)

[3] Bekka, M.E.B. and Valette, A.: “Group cohomology, harmonic functions and the first L^{2}-Betti number”, preprint, to appear in Potential Analysis (1994)

[4] Brown, K.S.: “Cohomology of groups”, Graduate Texts in Mathematics 87, Springer (1982)

[5] Brown, K.A. and Lorenz, M.: “Colimits of functors, and Grothendieck groups of infinite group algebras”, in “Abelian groups and noncommutative rings”, collect. papers in Mem. of R.B. Warfield jun., Contemp. Math. 130, 89 - 109 (1992)

[6] Cheeger, J. and Gromov, M.: “L^{2}-cohomology and group cohomology”, Topology 25, 189 - 215 (1986)

[7] Davis, J. and Lück, W.: “Spaces over a category and assembly maps in isomorphism conjectures in K-and L-Theory”, MPI-preprint, to appear in K-theory (1996)

[8] tom Dieck, T.: “Transformation groups and representation theory”, Lecture notes in mathematics 76, Springer (1979)

[9] tom Dieck, T.: “Transformation groups”, Studies in Math. 8, de Gruyter (1987)

[10] Eckmann, B.: “Amenable groups and Euler characeristics.”, Comm. Math. Helv. 67, 383 - 393 (1992)

[11] Eckmann, B.: “4-manifolds, group invariants and l^{2}-Betti numbers” preprint, to appear in L’Enseignement Mathématique (1997)

[12] Farber, M.S.: “Homological algebra of Novikov-Shubin invariants and Morse inequalities”, Geometric Analysis and Functional Analysis 6, 628 - 665 (1996)

[13] Kadison, R.V. and Ringrose, J.R.: “Fundamentals of the theory of operator algebras, volume II : Advanced theory”, Pure and Applied Mathematics, Academic Press (1986)

[14] Linnell, P.: “Division rings and group von Neumann algebras”, Forum Math. 5, 561 - 576 (1993)

[15] Lott, J.: “The Zero-in-the-Spectrum Question”, to appear in L’Enseignement Mathématique preprint (1997)

[16] Lott, J. and Lück, W.: “L^{2}-topological invariants of 3-manifolds”, Inventiones Math. 120, 15 -60 (1995)

[17] Lück, W.: “Transformation groups and algebraic K-theory”, Lecture Notes in Mathematics 1408 (1989)

[18] Lück, W.: “L^{2}-Betti numbers of mapping tori and groups”, Topology 33, 203 - 214 (1994)

[19] Lück, W.: “Hilbert modules and modules over finite von Neumann algebras and applications to L^{2}-invariants”, preprint, to appear in Math. Annalen (1995)

[20] Lück, W.: “L^{2}-invariants of regular coverings of compact manifolds and CW - complexes”, to appear in handbook of geometry“, Elsevier (1998)

[21] Lück, W.: “Dimension theory of arbitrary modules over finite von Neumann algebras and L^{2}-Betti numbers I: Foundations”, to appear in Journal für reine und angewandte Mathematik, (1998)

[22] Mislin, G.: “The geometric realization of Wall obstructions by nilpotent and simple spaces”, Math. Proc. Camb. Phil. Soc. 87, 199 - 206 (1980)

[23] Moody, J. A.: “Brauer induction for G_{0} of certain infinite groups”, Journal of Algebra 122, 1–14 (1989)

[24] Rowen, L.: “Ring Theory, Volume II”, Pure and Applied Mathematics 128, Academic Press (1988)

[25] Wall, C.T.C.: “Rational Euler characteristics”, Proc. Camb. Phil. Soc. 51, 181 - 183 (1961)

[26] Wall, C.T.C.: “Finiteness conditions for CW -complexes”, Annals of Mathematics 81, 59–69 (1965)

[27] Weibel, C.: “An introduction to homological algebra”, Cambridge University Press (1994)

[28] Whitehead, G.W.: “Elements of homotopy theory”, Graduate Texts in Mathematics, Springer (1978)

Current address Wolfgang Lück Fachbereich Mathematik und Informatik Westfälische Wilhelms-Universität Münster Einsteinstr. 62 48149 Münster Bundesrepublik Deutschland email: lueck@math.uni-muenster.de FAX: 0251 8338370 internet: http://wwwmath.uni-muenster.de/math/u/lueck/

Version of September 30, 1997