A Panorama on L^{2}-Invariants

and Their Applications

Wolfgang Lück^{∗}

Fachbereich Mathematik und Informatik

Westfälische Wilhelms-Universität

Münster Einsteinstr. 62 48149 Münster

Germany

lueck@math.uni-muenster.de http://www.math.unimuenster.de/u/lueck

May 2004

1. L^{2}-Betti Numbers

Principle 1.1 Given an invariant for finite CW -complexes, one can get much more sophisticated versions by passing to the universal covering and defining an analogue taking the action of the fundamental group into account.

Examples:

Classical notion generalized version Homology with ℤcoefficients Homology with coefficients in representations

Euler characteristic ∈ ℤ Walls finiteness obstruction in K_{0}(ℤπ)

Lefschetz numbers ∈ ℤ Generalized Lefschetz invariants in ℤπ_{ϕ}

Signature ∈ ℤ Surgery invariants

in L_{∗}(ℤG)

— torsion invariants Goal 1.2 Apply this principle to (classical) Betti numbers

b_{p}(X) = dim_{ℂ}(H_{p}(X; ℂ)).

Here are two naive attempts which fail:

• dim_{ℂ}(H_{p}(X̃; ℂ))

• dim_{ℂπ}(H_{p}(X̃; ℂ)), where dim_{ℂπ}(M) for a ℂ[π]-module could be choosen as dim_{ℂ}(ℂ ⊗_{ℂG} M).

We will use the following successful approach essentially due to Atiyah:

Throughout these lectures let G be a discrete group.

Denote by l^{2}(G) the Hilbert space of (formal) sums ∑_{g∈G} λ_{g} ·g such that λ_{g} ∈ ℂ and ∑_{g∈G} |λ_{g}|^{2} < ∞.

Definition 1.3 Define the group von Neumann algebra

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

to be the algebra of bounded G-equivariant operators l^{2}(G) → l^{2}(G).

The von Neumann trace is defined by

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

Example 1.4 If G is finite, then ℂG = l^{2}(G) = N (G). The trace tr_{N (G)} assigns to ∑_{g∈G} λ_{g} · g the coefficient λ_{e}.

Example 1.5 Let G be ℤ^{n}.

Let L^{2}(T^{n}) be the Hilbert space of L^{2}- integrable functions T^{n} → ℂ.

Let L^{∞}(T^{n}) be the Banach space of essentially bounded functions f: T^{n} → ℂ ∐{∞}. 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}).

We obtain an isomorphism

L^{∞}(T^{n}) ⟶^{≅} N (ℤ^{n}) = B(L^{2}(T^{n}))^{ℤ^{n}}

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

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

Under this identification the trace becomes

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

Definition 1.6 A finitely generated Hilbert N (G)-module V is a Hilbert space V together with a linear isometric G-action such that there exists an isometric linear G-embedding of V into l^{2}(G)^{n} for some n ≥ 0.

A map of finitely generated Hilbert N (G)- modules f: V → W is a bounded G-equivariant operator.

Definition 1.7 Let V be a finitely generated Hilbert N (G)-module. Choose a Gequivariant projection p: l^{2}(G)^{n} → l^{2}(G)^{n} with im(p) ≅_{N (G)} V. Define the von Neumann dimension of V by

dim_{N (G)}(V) := tr_{N (G)}(p)

:= n ; ∑ ; i=1 tr_{N (G)}(p_{i,i}) ∈ [0, ∞).

Example 1.8 For finite G a finitely generated Hilbert N (G)-module V is the same as a unitary finite dimensional G-representation and

dim_{N (G)}(V) = \frac{1}{|G|} · dim_{ℂ}(V).

Example 1.9 Let G be ℤ^{n}. Let X ⊂ T^{n} be any measurable set with characteristic function χ_{X} ∈ L^{∞}(T^{n}). Let M_{χ_{X}}: L^{2}(T^{n}) → L^{2}(T^{n}) be the ℤ^{n}-equivariant unitary projection given by multiplication with χ_{X}. Its image V is a Hilbert N (ℤ^{n})-module with

dim_{N (ℤ^{n})}(V) = vol(X).

In particular each r ∈ [0, ∞) occurs as r = dim_{N (ℤ^{n})}(V).

Definition 1.10 A sequence of Hilbert N (G)- modules U ⟶^{i} V ⟶^{p} W is weakly exact at V if the kernel ker (p) of p and the closure clos (im(i)) of the image im(i) of i agree.

A map of Hilbert N (G)-modules f: V → W is a weak isomorphism if it is injective and has dense image.

Example 1.11 The morphism of N (ℤ)-Hilbert modules

M_{z−1} : l^{2}(ℤ) → l^{2}(ℤ), u ↦ (z − 1) · u

is a weak isomorphism, but not an isomorphism.

Theorem 1.12 1. Faithfulness We have for a Hilbert N (G)-module V

V = 0 ⟺ dim_{N (G)}(V) = 0;

2. Additivity If 0 → U → V → W → 0 is a weakly exact sequence of Hilbert N (G)-modules, then

dim_{N (G)}(U) + dim_{N (G)}(W) = dim_{N (G)}(V);

3. Cofinality Let {V_{i} | i ∈ I} be a directed system of Hilbert N (G)- submodules of V, directed by ⊂. Then

dim_{N (G)}(clos (∪_{i∈I}V_{i}))

= sup {dim_{N (G)}(V_{i}) | i ∈ I}; Definition 1.13 A G-CW -complex X is a G-space 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 and X_{n} is obtained from X_{n−1} 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}

We call X finite if it is built by finitely many equivariant cells. We call X of finite type if each skeleton X_{n} is finite.

Definition 1.14 Let X be a free G-CW - complex of finite type. Denote by C_{∗}(X) its cellular ℤG-chain complex. Define its cellular L^{2}-chain complex C_{∗}^{(2)} (X) to be the Hilbert N (G)-chain complex

C_{∗}^{(2)} (X) := l^{2}(G) ⊗_{ℤG} C_{∗}(X).

Define its p-th L^{2}-homology to be the finitely generated Hilbert N (G)-module

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

Define its p-th L^{2}-Betti number

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

Remark 1.15 Notice that C_{p}(X) = ⊕_{I_{p}}ℤG. Hence C_{p}^{(2)} (X) = ⊕_{I_{p}}l^{2}(G). In particular

dim_{N (G)}(C_{p}^{(2)} (X)) = |I_{p}| = |{p-cells in G\X}|.

Each differential c_{p}^{(2)} is a morphism of finitely generated Hilbert N (G)-modules since each I_{p} is finite by assumption.

Theorem 1.16 (Cellular L^{2}-Betti numbers)

1. Homotopy invariance Let f: X → Y be a G-map of free G-CW -complexes of finite type. If f is a weak homotopy equivalence (after forgetting the G-action), then

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

2. Euler-Poincaré formula, Atiyah Let X be free finite G-CW -complex. Let χ(G\X) be the Euler characteristic of the finite CW -complex G\X. Then

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

3. Poincaré duality Let M be a cocompact free proper Gmanifold of dimension n which is orientable. Then

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

4. Künneth formula, Zucker Let X be a free G-CW -complex of finite type and Y be a free H-CW -complex of finite type. Then we get for all n ≥ 0

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

5. Wedges Let X_{1}, X_{2}, …, X_{r} be connected 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 + r ; ∑ ; j=1 (b_{1}^{(2)} (X̃_{j}) − b_{0}^{(2)} (X̃_{j})) ;

and for 2 ≤ p

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

6. Morse inequalities, Novikov-Shubin Let X be a free G-CW -complex of finite type. Then we get for n ≥ 0

n ; ∑ ; p=0 (−1)^{n−p} · b_{p}^{(2)} (X)

≤ n ; ∑ ; p=0 (−1)^{n−p} · |{p-cells of G\X)}|;

7. Zero-th L^{2}-Betti number Let X be a connected free G-CW -complex of finite type. Then

b_{0}^{(2)} (X) = \frac{1}{|G|};

8. Restriction Let X be a free G-CW -complex of finite type and let H ⊂ G be a subgroup of finite index [G : H]. Then

[G : H] · b_{p}^{(2)} (X; N (G)) = b_{p}^{(2)} (res _{G}^{H} X; N (H)); 9. Induction

Let H ⊆ G and let X be a free H-CW - complex of finite type. Then

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

Example 1.17 If G is finite and X is a free G-CW -complex of finite type, then b_{p}^{(2)} (X) is the classical p-th Betti number of X multiplied with \frac{1}{|G|}.

Example 1.18 Consider the ℤ-CW -complex S̃^{1}. We get for C_{∗}^{(2)} (S̃^{1})

… → 0 → l^{2}(ℤ) ⟶^{M_{z−1}} l^{2}(ℤ) → 0 → …

and hence H_{p}^{(2)} (S̃^{1}) = 0 and b_{p}^{(2)} (S̃^{1}) = 0 for all p.

Example 1.19 Let F_{g} be the orientable closed surface. For F_{0} = S^{2} we get

b_{p}^{(2)} (F̃_{0}) = b_{p}(S^{2}) = \{ \begin{matrix}1 & if p = 0, 2; \\ 0 & otherwise. \end{matrix}

If g ≥ 0, then |π_{1}(F_{g})| = ∞ and hence b_{0}^{(2)} (F̃_{g}) = 0. By Poincaré duality b_{2}^{(2)} (F̃_{g}) =

0. As dim (F_{g}) = 2, we get b_{p}^{(2)} (F̃_{g}) = 0 for p ≥ 3. The Euler-Poincaré formula shows

b_{1}^{(2)} (F̃_{g}) = −χ(F_{g}) = 2g − 2;
b_{p}^{(2)} (F̃_{0}) = 0 for p ≠ 1.

Example 1.20 Let X → Y be a covering with d-sheets of connected CW -complexes of finite type. Theorem 1.16 (8) implies

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

In particular we get for a connected CW - complex X of finite type, for which there is a selfcovering X → X with d-sheets for some integer d ≥ 2, that for p ≥ 0

b_{p}^{(2)} (X̃) = 0.

This implies for each finite CW -complex Y of finite type

b_{p}^{(2)} ({ \atop S^{1} × Y}) = 0.

Theorem 1.21 (Vanishing of L^{2}-Betti numbers of mapping tori, L.) Let f: X → X be a cellular selfhomotopy equivalence of a connected CW -complex X of finite type. Then we get for all p ≥ 0

b_{p}^{(2)} (T̃_{f}) = 0.

Proof: As T_{f^{d}} → T_{f} has d sheets, we get

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

If β_{p}(X) is the number of p-cells, then there is up to homotopy equivalence a CW - structure on T_{f^{d}} with β(T_{f^{d}}) = β_{p}(X) + β_{p−1}(X). We have

b_{p}^{(2)} (T̃_{f^{d}}) = dim_{N (G)} (H_{p}^{(2)} (C_{p}^{(2)} (T̃_{f^{d}}))

≤ dim_{N (G)} (C_{p}^{(2)} (T̃_{f^{d}})) = β_{p}(T_{f^{d}}).

This implies for all d ≥ 1

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

Taking the limit for d → ∞ yields the claim. □

2. Further Results about L^{2}-Betti Numbers

Theorem 2.1 (Long weakly exact L^{2}- homology sequence, Cheeger-Gromov)

Let 0 → C_{∗} ⟶^{i_{∗}} D_{∗} ⟶^{p_{∗}} E_{∗} → 0 be an ex-

act sequence of chain complexes of finitely generated Hilbert N (G)-modules. Then there is a long weakly exact homology sequence

… ⟶^{H_{n+1}^{(2)} (p_{∗})} H_{n+1}^{(2)} (E_{∗}) ⟶^{∂_{n+1}} H_{n}^{(2)} (C_{∗})

⟶^{H_{n}^{(2)} (i_{∗})} H_{n}^{(2)} (D_{∗}) ⟶^{H_{n}^{(2)} (p_{∗})} H_{n}^{(2)} (E_{∗}) ⟶^{∂_{n}} ….

Theorem 2.2 (L^{2}-Betti numbers and S^{1}- actions, L.) Let X be a connected S^{1}-CW -complex of finite type, for instance a connected compact manifold with S^{1}-action. Suppose that for one (and hence all) x ∈ X the map S^{1} → X, z ↦ zx is π_{1}-injective. (In particular the S^{1}-action has no fixed points.) Then we get for all p ≥ 0

b_{p}^{(2)} (X̃) = 0.

Theorem 2.3 (L^{2}-Betti numbers and aspherical S^{1}-manifolds, L.) Let M be an aspherical closed manifold with non-trivial S^{1}-action. Then

1. The action has no fixed points;

2. The map S^{1} → X, z ↦ zx is π_{1}-injective for x ∈ X;

3. b_{p}^{(2)} (M̃) = 0 for p ≥ 0 and χ(M) = 0.

Theorem 2.4 (L^{2}-Hodge-de Rham Theorem, Dodziuk) Let M be a cocompact free proper Gmanifold with G-invariant Riemannian metric and ∂M = ∅. Let K be an equivariant smooth triangulation of M. Put

H_{(2)}^{p} (M) = {ω ∈ Ω^{p}(M) | Δ_{p}(ω) = 0, ||ω||_{L^{2}} < ∞}.

Then integration defines an isomorphism of finitely generated Hilbert N (G)-modules

H_{(2)}^{p} (M) ⟶^{≅} H_{(2)}^{p} (K).

Corollary 2.5

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

where F is a fundamental domain for the G-action and e^{−tΔ_{p}} (x, y) is the heat kernel on M̃.

Theorem 2.6 (Dodziuk) 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}

Proof: A direct computation shows that H_{(2)}^{p} (H^{n}) is not zero if and only if 2p = n. Notice that M is hyperbolic if and only if M̃ is isometrically diffeomorphic to the standard hyperbolic space H^{n}. □

Corollary 2.7 Let M be a hyperbolic closed manifold of dimension n. Then

1. If n = 2m is even, then

(−1)^{m} · χ(M) > 0;

2. M carries no non-trivial S^{1}-action.

Proof: (1) We get from the Euler-Poincaré formula and Theorem 2.6

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

(2) We give the proof only for n = 2m even. Then b_{m}^{(2)} (M̃) > 0. Since M̃ = H^{n} is contractible, M is aspherical. Now apply Theorem 2.3. □

Theorem 2.8 (L^{2}-Betti numbers of 3manifolds, Lott-L.) 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

b_{0}^{(2)} (M̃) = 0;
b_{1}^{(2)} (M̃) = (r − 1) − r ; ∑ ; j=1 \frac{1}{| π_{1}(M_{j}) |} − χ(M)

+ |{C ∈ π_{0}(∂M) | C ≅ S^{2}}| ;

b_{2}^{(2)} (M̃) = (r − 1) − r ; ∑ ; j=1 \frac{1}{| π_{1}(M_{j}) |}

+ |{C ∈ π_{0}(∂M) | C ≅ S^{2}}| ;

b_{3}^{(2)} (M̃) = 0.

Lemma 2.9 Let X be a free ℤ^{n}-CW -complex of finite type. Then

b_{p}^{(2)} (X) = dim_{ℂ[ℤ^{n}]^{(0)}} (ℂ[ℤ^{n}]^{(0)} ⊗_{ℤ[ℤ^{n}]} H_{p}(X)) ,

where ℂ[ℤ^{n}]^{(0)} is the quotient field of ℂ[ℤ^{n}].

Example 2.10 In general there are no relations between the Betti numbers b_{p}(X) and the L^{2}-Betti numbers b_{p}^{(2)} (X̃) for a connected CW -complex X of finite type.

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)} (BG) = r_{p} for 1 ≤ p ≤ l;
b_{p}^{(2)} (BG) = 0 for l + 1 ≤ p;
b_{p}(BG) = 0 for p ≥ 1.

Namely, take for appropriate k, l, m_{i}, n_{i}

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

G_{i}(m_{i}, n_{i}) = ℤ/n_{i}×(∗_{k=1}^{2m_{i}+2} ℤ/2)×( i−1 ; ∏ ; j=1 ∗_{l=1}^{4} ℤ/2) .

For any sequence n_{1}, n_{2}, … of non-negative integers there is a CW -complex X of finite type such that for p ≥ 1

b_{p}(X) = n_{p};
b_{p}^{(2)} (X̃) = 0.

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

Theorem 2.11 Approximation Theorem, L.) Let X be a free G-CW -complex of finite type. Suppose that G is residually finite, i.e. there is a nested sequence

G = G_{0} ⊃ G_{1} ⊃ G_{2} ⊃ …

of normal subgroups of finite index with ∩_{n≥1}G_{n} = {1}. Then for any such sequence (G_{n})_{n≥1}

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

Remark 2.12 Ordinary Betti numbers are not multiplicative under finite coverings, whereas the L^{2}-Betti numbers are, i.e. for a d-sheeted covering p: X → Y we get

b_{p}^{(2)} (X̃; N (π_{1}(X)) = d · b_{p}^{(2)} (Ỹ; N (π_{1}(Y)).

With the expression lim_{n→∞} \frac{b_{p}(G\X)}{[G:G_{n}]} we try to force the Betti numbers to be multiplicative by a limit process.

Theorem 2.11 says that L^{2}-Betti numbers are asymptotic Betti numbers. It was conjectured by Gromov.

3. L^{2}-Torsion

There is another interesting L^{2}-invariant, the L^{2}-torsion

ρ^{(2)} (X̃) ∈ ℝ.  (3.1)

It is defined for a finite connected CW - complex X such that X̃ is L^{2}-acyclic i.e. b_{p}^{(2)} (X̃) = 0 for p ≥ 0. (We ignore questions about determinant class).

Theorem 3.2 (Cellular L^{2}-torsion for universal coverings, L.)

1. 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. Suppose that X̃ or Ỹ is L^{2}- acyclic. Then both X̃ and Ỹ are 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)) → ℝ is given by the Fuglede-Kadison determinant; 2. Sum formula

Consider the cellular pushout of finite CW -complexes

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

Assume X̃_{0}, X̃_{1}, and X̃_{2} are L^{2}-acyclic and that for k = 0, 1, 2 the obvious map i_{k}: X_{k} → X are π_{1}-injective.

Then X̃ is L^{2}-acyclic and we get

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

3. Poincaré duality Let M be a closed manifold of even dimension such that M̃ is L^{2}-acyclic. Then

ρ^{(2)} (M̃) = 0;

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

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

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

ρ^{(2)} (X̃) = d · ρ^{(2)} (Ỹ);

Remark 3.3 Notice the formal analogy between the behaviour of ρ^{(2)} (X̃) and the ordinary Euler characteristic χ(X).

The next result follows from work of Burghelea-Friedlander-Kappeler-McDonald and Schick-L.

Theorem 3.4 Let M be a compact irreducible 3-manifold with infinite fundamental group such that its boundary is empty or a disjoint union of incompressible tori. Suppose that M satisfies Thurston’s Geometrization Conjecture. Let M_{1}, M_{2}, …, M_{r} be the hyperbolic pieces in the Jaco-Shalen-Johannson-Thurston splitting along incompressible embedded tori. Then

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

Theorem 3.5 (Hess-Schick) The L^{2}-torsion ρ^{(2)} (M̃) for a closed hyperbolic manifold M of odd dimension is up to a (computable) non-zero dimension constant equal to vol(M).

Definition 3.6 (L^{2}-torsion of group automorphisms) Let f: G → G be a group automorphism. Suppose that there is a finite CW -model for BG. 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 3.7 (L^{2}-torsion of group automorphisms, L.). Suppose that all groups appearing below have finite classifying spaces.

1. Amalgamated Products 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}); 2. Trace property

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;

3. Additivity 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});

4. Multiplicativity Let f: G → G be an automorphism of a group. Then for all integers n ≥ 1

ρ^{(2)} (f^{n}) = n · ρ^{(2)} (f);

5. Subgroups of finite index 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}});

6. Dependence on L^{2}-homology Let f: G → G be an automorphism of a group G. Then ρ^{(2)} (f) depends only on the map H_{p}^{(2)} (B̃f): H_{p}^{(2)} (B̃G) → H_{p}^{(2)} (B̃G) induced by f on the L^{2}-homology of the universal covering of BG.

7. Vanishing results We have ρ^{(2)} (f) = 0 if G satisfies one of the following conditions:

(a) All L^{2}-Betti numbers of the universal covering of BG vanish;

(b) G contains an amenable infinite normal subgroup.

Remark 3.8 Let f: S → S be an automorphisms of a compact connected orientable surface. Let M_{1}, M_{1}, …, M_{r} be the hyperbolic pieces in the JSJT-splitting. If S is S^{2}, D^{2}, or T^{2}, then ρ^{(2)} (f) = 0. Otherwise we get

ρ^{(2)} (π_{1}(f): π_{1}(S) → π_{1}(S)) = \frac{−1}{6π}· r ; ∑ ; i=1 vol(M_{i}).

Suppose f is irreducible. Then f is pseudo-Anosov if and only if ρ^{(2)} (f) < 0 and f is periodic if and only if ρ^{(2)} (f) = 0.

Question 3.9 Does for G a finitely generated free group or a surface group ρ^{(2)} (f) determine the conjugacy class of f up to finite ambiguity?

Theorem 3.10 (L^{2}-Betti numbers and fibrations, L.). Let F → E → B be a fibration of connected CW -complexes of finite type. Suppose that π_{1}(F) → π_{1}(E) is injective and b_{p}^{(2)} (F̃) = 0 for all p ≥ 0. Then b_{p}^{(2)} (Ẽ) = 0 for all p ≥ 0.

Theorem 3.11 (Proportionality Principle for L^{2}-Betti numbers, Cheeger-Gromov) 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 such that for any discrete group G with a free proper cocompact action on M by isometries the following holds

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

The analogous statement holds for the L^{2}- torsion

4. The Generalized Dimension function

In this section we present a purely algebraic approach to L^{2}-Betti numbers.

Remark 4.1 Recall that by definition

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

This induces a bijection of ℂ-vector spaces

M(m, n, N (G)) ⟶^{≅} mor _{N (G)}(l^{2}(G)^{m}, l^{2}(G)^{n}).

It is compatible with multiplication of matrices and composition of morphisms. This extends to finitely generated Hilbert N (G)- modules and finitely projective N (G)-modules.

Theorem 4.2 (Modules over N (G) and Hilbert N (G)-modules) We obtain an equivalence of ℂ-categories

ν: {fin. gen. proj. N (G)-mod. } → {fin. gen. Hilb. N (G)-mod. }.

Definition 4.3 Let R be a ring. Let M be a 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 a R-module M define the R-submodule TM and the R-quotient module PM by:

TM := {x ∈ M | f(x) = 0

for all f ∈ M^{∗}};

PM := M/TM.

We call a sequence of R-modules L ⟶^{i} M ⟶^{q} N weakly exact if \overline{im(i)} = ker (q).

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 and that PM = 0 is equivalent to M^{∗} = 0.

Example 4.4 Let R = ℤ. Let M be a finitely generated ℤ-module and K ⊂ M. Then

\overline{K} = \{x ∈ M | n · x ∈ K for some n ∈ ℤ\};
TM := tors (M);
PM = M/ tors (M).

A sequence M_{0} → M_{1} → M_{2} of finitely generated ℤ-modules is weakly exact if and only if it is exact after applying ℚ ⊗_{ℤ} −.

Definition 4.5 Let P be a finitely generated projective N (G)-module. Choose a matrix A ∈ M_{n}(N (G)) with A^{2} = A such that the image of r_{A}: N (G)^{n} → N (G)^{n} is N (G)-isomorphic to P. Define

dim_{N (G)}(P) := tr_{N (G)}(A) [0, ∞).

Lemma 4.6 1. The functors ν and ν^{−1} preserve exact sequences and weakly exact sequences;

2. If P is a finitely generated projective N (G)-module, then

dim_{N (G)}(P) = dim_{N (G)}(ν(P)).

Remark 4.7 N (G) is Noetherian if and only if G is finite. It contains zero-divisors if G is non-trivial.

Definition 4.8 A ring R is called semihereditary if any finitely generated submodule of a projective module is projective.

Lemma 4.9 N (G) is semihereditary.

Proof: It suffices to prove for a finitely generated N (G)-submodule M ⊂ N (G)^{n} that it is projective. Choose a N (G)-map f: N (G)^{m} → N (G)^{n} whose image is M. Let ν(f): l^{2}(G)^{m} → l^{2}(G)^{n} be the morphism corresponding to f under ν. Choose a projection pr : l^{2}(G)^{m} → l^{2}(G)^{m} with image ker (ν(f)). Then

l^{2}(G)^{m} ⟶^{pr} l^{2}(G)^{m} ⟶^{ν^{−1}(f)} l^{2}(G)^{n}

is exact. Hence

N (G)^{m} ⟶^{ν^{−1}(pr)} N (G)^{m} ⟶^{f} N (G)^{n}

is exact and ν^{−1}(pr)^{2} = ν^{−1}(pr). Hence ker (f) ⊂ N (G)^{m} is a direct summand and M = im(f) is projective. □

Remark 4.10 The following results and definitions can be understood by the slogan that N (G) behaves like ℤ if one forgets that ℤ is Noetherian and has no-zero-divisors. In this sense all properties of ℤ carry over to N (G).

Lemma 4.11 Let M be a finitely generated N (G)-module. Then

1. Let K ⊂ M be a submodule. Then \overline{K} ⊂ M is a direct summand and M/\overline{K} is finitely generated projective;

2. PM is a finitely generated projective N (G)-module and we get a splitting

M ≅ TM ⊕ PM;

3. If M is finitely presented, then there is an exact sequence

0 → N (G)^{n} → N (G)^{n} → TM → 0.

Theorem 4.12 (Dimension function for arbitrary N (G)-modules, L.)

There is precisely one dimension function

dim : {N (G) − modules } → [0, ∞]

which has the following properties;

1. Extension Property If M is a finitely generated projective R-module, then dim (M) agrees with the previously defined notion;

2. Additivity If 0 → M_{0} ⟶^{i} M_{1} ⟶^{p} M_{2} → 0 is an exact sequence of R-modules, then

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

3. 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};

4. Continuity If K ⊂ M is a submodule of the finitely generated R-module M, then

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

5. Dimension and Torsion If M is a finitely generated R-module, then

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

Proof: We give the proof of uniqueness which leads to the definition of dim . Any N (G)-module M is the colimit over the directed system of its finitely generated submodules {M_{i} | i ∈ I}. Hence by Cofinality

dim (M) = sup {dim (M_{i}) | i ∈ I}.

We get for each M_{i} from Additivity

dim (M_{i}) = dim (PM_{i}).

Hence we get

dim (M) = sup {dim (P) | P ⊂ M finitely generated projective }.□

Definition 4.13 Let X be a (left) G-space. Its homology with coefficients in N (G) is

H_{p}^{G}(X; N (G)) = H_{p} (N (G) ⊗_{ℤG} C_{∗}^{sing} (X)) .

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, ∞].

Lemma 4.14 Let X be a free G-CW -complex of finite type. Then Definition 4.13 of L^{2}- Betti numbers b_{p}^{(2)} (X; N (G)) agrees with the previous one.

Definition 4.15 The p-th L^{2}-Betti number of a group G is

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

Theorem 4.16 L^{2}-Betti numbers for arbitrary spaces, L.)

1. Homotopy invariance Let f: X → Y be a G-map. Suppose such that for each subgroup H ⊂ G the induced map f^{H}: X^{H} → Y^{H} is a homology equivalence (for singular homology with ℂ-coefficients). Then for all p ≥ 0

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

2. 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)); 3. Künneth formula

Let X be a G-space and Y be a Hspace. 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 0 · ∞ := 0, r · ∞ := ∞ for r ∈ (0, ∞] and r + ∞ = ∞ for r ∈ [0, ∞];

4. Induction Let H ⊆ G be a subgroup. Then

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

5. Restriction Let H ⊂ G be a subgroup of finite index [G : H]. Let X be a G-space. Then

b_{p}^{(2)} (res (X); N (H))

= [G : H] · b_{p}^{(2)} (X; N (G));

6. Zero-th homology and L^{2}-Betti number For a path-connected G-space X

b_{0}^{(2)} (X; N (G)) = |G|^{−1}.

Definition 4.17 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

inf {f(g) | g ∈ G} ≤ μ(f) ≤ sup {f(g) | g ∈ G} for all f ∈ l^{∞}(G, ℝ).

The class of elementary amenable groups is defined as the smallest class of groups, which contains all finite and all abelian groups and is closed under i.) taking subgroups, ii) taking quotient groups iii.) under extensions and iv.) under directed unions.

Remark 4.18 The class of amenable groups contains the class of elementary amenable groups. A group which contains ℤ ∗ ℤ is not amenable.

Theorem 4.19 (Dimension-flatness of N (G) over ℂG for amenable G), L.) Let G be amenable and M be a ℂG-module. Then for p ≥ 1

dim_{N (G)} (Tor _{p}^{ℂG}(N (G), M)) = 0.

Theorem 4.20 Let G be an amenable group and X be a G-space. Then

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

Corollary 4.21 (Cheeger-Gromov) Let G be a group which contains an infinite normal amenable subgroup. Then for p ≥ 0

b_{p}^{(2)} (G; N (G)) = 0.

If there is a finite model for BG, then

χ(G) := χ(BG) = 0.

Proof: If G is amenable, this follows from H_{p}(EG; ℂ) = 0 for p ≥ 1. In the general case use a spectral sequence argument. □

Definition 4.22 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 R-modules 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 Rmodules. Define G_{0}(R) analogously but replacing finitely generated projective by finitely generated.

Theorem 4.23 (L.) Let G be an amenable group. Then we get a well-defined map

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

In particular [ℂG] generates an infinite cyclic subgroup in G_{0}(ℂG).

Lemma 4.24 If G contains ℤ ∗ ℤ as subgroup, then

[ℂG] = 0 ∈ G_{0}(ℂG).

Conjecture 4.25 G is amenable if and only if

[ℂG] ≠ 0 ∈ G_{0}(ℂG).

Remark 4.26 Elek has generalized Theorem 4.23 to arbitrary fields as coefficients instead of ℂ by defining dimension functions also in this context.

Definition 4.27 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.

Example 4.28 The free group F_{g} has the obvious presentation ⟨s_{1}, s_{2}, … s_{g} | ∅⟩ and its deficiency is realized by this presentation, namely def (F_{g}) = g.

If G is a finite group, def (G) ≤ 0 by Lemma 4.30 as b_{0}^{(2)} (G) = |G|^{−1} and b_{1}^{(2)} (G) = 0.

The deficiency of a cyclic group ℤ/n is 0, the obvious presentation ⟨s | s^{n}⟩ realizes the deficiency.

The deficiency of ℤ/n × ℤ/n is −1, the obvious presentation ⟨s, t | s^{n}, t^{n}, [s, t]⟩ realizes the deficiency.

Example 4.29 The deficiency is not additive under free products by the following example due to Hog, Lustig and Metzler(1985). 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}⟩.

Lemma 4.30 Let G be a finitely presented group. Then

def (G) ≤ 1 − b_{0}^{(2)} (G) + b_{1}^{(2)} (G) − b_{2}^{(2)} (G).

Proof We have to show for any presentation P that

g(P)−r(P) ≤ 1−b_{0}^{(2)} (G)+b_{1}^{(2)} (G)−b_{2}^{(2)} (G).

Let X be a CW -complex realizing P. Then

χ(X) = 1 − g(P) + r(P) = b_{0}^{(2)} (X̃) + b_{1}^{(2)} (X̃) − b_{2}^{(2)} (X̃).

Since the classifying map X → BG is 2connected, we get

b_{p}^{(2)} (X̃) = b_{p}^{(2)} (G) for p = 0, 1;
b_{2}^{(2)} (X̃) ≥ b_{2}^{(2)} (G). □

Theorem 4.31 (Deficiency and extensions, L.) 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.

1. b_{1}^{(2)} (H) < ∞;

2. The ordinary first Betti number of H satisfies b_{1}(H) < ∞ and b_{1}^{(2)} (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).

Let F be Thompson’s group. It is the group of orientation preserving dyadic PLautomorphisms 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⟩.

It has a model of finite type for BG but no finite-dimensional model since ℤ^{n} is a subgroup of F for all n ≥ 1. It does not contain ℤ ∗ ℤ as subgroup and is not elementary amenable. It is not known whether it is amenable.

Theorem 4.32 (L.) The L^{2}-Betti numbers of Thompson’s group b_{p}^{(2)} (BF) vanish for all p ≥ 0.

Theorem 4.33 (L^{2}-Betti numbers and S^{1}-actions, L.) 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.) Let X̃ be the universal covering of X with the canonical π_{1}(X)-action. Then we get for all p ≥ 0

b_{p}^{(2)} (X̃) = 0.

Theorem 4.34 (L^{2}-Betti numbers and fibrations, L.) Let F ⟶^{i} E ⟶^{p} B be a fibration of connected CW -complexes. Suppose that π_{1}(F) → π_{1}(E) is injective. Suppose for a given integer d ≥ 1 that b_{p}^{(2)} (F̃) = 0 for p ≤ d−1 and b_{d}^{(2)} (F̃) < ∞ holds. Suppose that π_{1}(B) contains an element of infinite order or finite subgroups of arbitrary large order. Then b_{p}^{(2)} (Ẽ) = 0 for p ≤ d.

5. Survey on Further Results and Conjectures

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

\frac{1}{|FIN (G)|} ℤ ⊂ ℚ

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

Conjecture 5.1 (Atiyah Conjecture) A group G satisfies the Atiyah Conjecture if for any matrix A ∈ M(m, n, ℚG) the von Neumann dimension of the kernel of the G-equivariant bounded operator r_{A}^{(2)} : l^{2}(G)^{m} → l^{2}(G)^{n}, x ↦ xA satisfies

dim_{N (G)} (ker (r_{A}^{(2)} : l^{2}(G)^{m} → l^{2}(G)^{n}) )

∈ \frac{1}{|FIN (G)|} ℤ.

Remark 5.2 If G is torsionfree, then

\frac{1}{|FIN (G)|} ℤ = ℤ.

Lemma 5.3 Let G be a group. Then the following statements are equivalent:

1. For any cocompact free proper G-manifold M without boundary we have

b_{p}^{(2)} (M; N (G)) ∈ \frac{1}{|FIN (G)|} ℤ;

2. For any cocompact free proper G-CW - complex X we have

b_{p}^{(2)} (X; N (G)) ∈ \frac{1}{|FIN (G)|} ℤ;

3. For any finitely presented ℚG-module M

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

4. The Atiyah Conjecture 5.1 is true for G.

Remark 5.4 Atiyah asked originally the following question. Let G → \overline{M} → M be a Gcovering of a closed Riemannian manifold M. Is then

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

a rational number?

Remark 5.5 The Farrell-Jones Conjecture for K_{0}(ℚG) says that the canonical map

colim _{H⊆G,|H|<∞} K_{0}(ℚH) → K_{0}(ℚG)

is bijective. Surjectivity of this map implies for any finitely generated projective ℚGmodule M that

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

The Atiyah-Conjecture requires this for all finitely presented ℂG-modules.

Conjecture 5.6 Kaplanski Conjecture) The Kaplanski Conjecture for a torsionfree group G and a field F says that the group ring F G has no non-trivial zero-divisors.

Lemma 5.7 The Kaplanski Conjecture holds for G and the field ℚ if the Atiyah Conjecture 5.1 holds for G.

Proof: Let x ∈ ℚG be a zero-divisor. Let r_{x}^{(2)} : l^{2}(G) → l^{2}(G) be given by right multiplication with x. We get

0 < dim_{N (G)} (ker (r_{x}^{(2)} )) ≤ 1

Since by assumption dim_{N (G)} (ker (r_{x}^{(2)} )) ∈ ℤ, we conclude

dim_{N (G)} (ker (r_{x}^{(2)} )) = 1.

Since ker (r_{x}^{(2)} ) is closed in l^{2}(G), we conclude ker (r_{x}^{(2)} ) = l^{2}(G) and hence x = 0. □ Definition 5.8 Let C be the smallest class of groups which contains all free groups and is closed under directed union and extensions with elementary amenable quotients.

Theorem 5.9 (Linnell) Let G be a group such that there is an upper bound on the orders of finite subgroups and G belongs to C. Then the Atiyah Conjecture 5.1 holds for G.

Remark 5.10 Schick has enlarged the class of torsionfree groups, for which the Atiyah Conjecture is true, considerably by approximations techniques. For instance the Atiyah Conjecture is known for all torsionfree groups which are residually elementary amenable torsionfree.

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.Then {e_{0}t, t} is a set of generators for L. The associate 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.

Theorem 5.11 Grigorchuk-Linnell-Schick-Żuk 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 5.1.

Remark 5.12 No counterexample to the Atiyah Conjecture 5.1 is known if one replaces \frac{1}{|FIN (G)|} ℤ by ℚ or if one assumes that there is a bound on the orders of finite subgroups of G.

Conjecture 5.13 (Singer Conjecture) If M is an aspherical closed manifold, then

b_{p}^{(2)} (M̃) = 0 if 2p ≠ dim (M).

If M is a closed 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}

Because of the Euler-Poincaré formula

χ(M) = _{p≥0}^{∑}(−1)^{p} · b_{p}^{(2)} (M̃)

the Singer Conjecture 5.13 implies the following conjecture provided that M is aspherical or has negative sectional curvature.

Conjecture 5.14 (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 for ϵ = (−1)^{dim (M)/2}

ϵ · χ(M) > 0  if sec (M) < 0;
ϵ · 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.

Remark 5.15 Under certain negative pinching conditions the Singer Conjecture has been proved by Ballmann-Brüning, Donelly-Xavier, Jost-Xin. Direct computations show that the Singer Conjecture 5.13 holds for a closed Riemannian manifold M if dim (M) ≤ 3 (assuming Thurston’s Geometrization) or if M is a locally symmetric space or if M carries an S^{1}-action.

Definition 5.16 A Kähler hyperbolic manifold is a closed connected Kähler manifold M whose fundamental form ω is d̃(bounded), i.e. its lift ω̃ ∈ Ω^{2}(M̃) to the universal covering can be written as d(η) holds for some bounded 1-form η ∈ Ω^{1}(M̃).

Theorem 5.17 (Gromov) Let M be a closed 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;

Remark 5.18 Let M be a closed Kähler manifold. It is Kähler hyperbolic if it admits some Riemannian metric with negative sectional curvature, or, if, generally π_{1}(M) is word-hyperbolic and π_{2}(M) is trivial. A consequence of Theorem 5.17 is that any Kähler hyperbolic manifold manifold is a projective algebraic variety Let X be a topological space and let C_{∗}^{sing} (X; ℝ) be its singular chain complex with real coefficients. Let S_{p}(X) be the set of all singular p-simplices. Then C_{p}(X; ℝ) is the real vector space with S_{p}(X) as basis. Define the L^{1}-norm of an element x ∈ C_{p}(X), which is given by the (finite) sum ∑_{σ∈S_{p}(X)} λ_{σ}· σ, by

||x||_{1} := _{σ}^{∑} |λ_{σ}|.

Define the L^{1}-seminorm of an element y in the p-th singular homology H_{p}^{sing} (X; ℝ) := H_{p}(C_{∗}^{sing} (X; ℝ)) by

||y||_{1} := inf {||x||_{1} | x ∈ C_{p}^{sing} (X; ℝ),
∂_{p}(x) = 0, y = [x]}.

Notice that ||y||_{1} defines only a semi-norm on H_{p}^{sing} (X; ℝ), it is possible that ||y||_{1} = 0 but y ≠ 0. The next definition is due to Gromov and Thurston.

Definition 5.19 Let M be a closed connected orientable manifold of dimension n. Define its simplicial volume to be the non-negative real number

||M|| := ||j([M])||_{1} ∈ [0, ∞)

for any choice of fundamental class [M] ∈ H_{n}^{sing} (M, ℤ) and j: H_{n}^{sing} (M; ℤ) → H_{n}^{sing} (M; ℝ) the change of coefficients map associated to the inclusion ℤ → ℝ.

Theorem 5.20 (Simplical volume of hyperbolic manifolds) Let M be a closed hyperbolic orientable manifold of dimension n. Let v_{n} be the volume of the regular ideal simplex in \overline{ℍ^{n}}. Then

||M|| = \frac{vol(M)}{v_{n}} .

Example 5.21 We have ||S^{2}|| = ||T^{2}|| =

0. Let F_{g} be the closed connected orientable surface of genus g ≥ 1. Then

||F_{g}|| = 2 · |χ(F_{g})| = 4g − 4.

Definition 5.22 Let M be a smooth manifold. Define its minimal volume minvol (M) to be the infimum over all volumes vol(M, g), where g runs though all complete Riemannian metrics on M, for which the sectional curvature satisfies | sec (M, g)| ≤ 1.

Example 5.23 Obviously any closed flat Riemannian manifold has vanishing minimal volume. Hence we get

minvol (T^{n}) = ||T^{n}|| = 0.

Let F_{g} be the closed orientable surface of genus g ≥ 2, then

minvol (F_{g}) = 2π · |χ(F_{g})| = 2π · |2 − 2g| = π · ||F_{g}||

essentially by the Gauss-Bonnet formula.

Notice that ||S^{2}|| = 0 and minvol (S^{2}) ≠ 0.

We have

minvol (ℝ^{2}) = 2π(1 + √2);
minvol (ℝ^{n}) = 0 for n ≥ 3.

Theorem 5.24 (Gromov-Thurston) Let M be a closed connected orientable Riemannian manifold of dimension n. Then

||M|| ≤ (n − 1)^{n} · n! · minvol (M).

Conjecture 5.25 (Simplical volume and L^{2}-invariants) Let M be an aspherical closed oriented manifold of dimension ≥ 1. Suppose that its simplicial volume ||M|| vanishes. Then

b_{p}^{(2)} (M̃) = 0 for p ≥ 0;
ρ^{(2)} (M̃) = 0.

Example 5.26 Let M be an aspherical closed orientable manifold. Then Conjecture 5.25 is true in the following cases:

• S^{1}-actions If M carries an S^{1}-action, then

minvol (M) = 0;
||M|| = 0;
b_{p}^{(2)} (M̃) = 0 for all p ≥ 0;

• Amenable normal non-trivial subgroups Let H ⊂ π_{1}(M) be a normal infinite subgroup. If H is amenable, then

||M|| = 0;
b_{p}^{(2)} (M̃) = 0 for all p ≥ 0;

If H is elementary amenable, then

ρ^{(2)} (M̃) = 0;

• Selfmaps of degree ≠ −1, 0, 1 If there is a selfmap f: M → M of degree deg (f) different from −1, 0, and 1, then

||M|| = 0.

.

If any normal subgroup of finite index of π_{1}(M) is Hopfian and there is a selfmap f: M → M of degree deg (f) different from −1, 0, and 1, then for all p ≥ 0

b_{p}^{(2)} (M̃) = 0;

• Dimension 3 Suppose that M has dimension 3 (and satisfies Thurston’s Geometrization Conjecture). Then ||M|| = 0 implies

b_{p}^{(2)} (M̃) = 0 for all p ≥ 0;
ρ^{(2)} (M̃) = 0;

• Vanishing of mininal volume If the minimal volume minvol (M) of M is zero, then

||M|| = 0;
b_{p}^{(2)} (M̃) = 0 for all p ≥ 0.

Definition 5.27 Two countable groups G_{0} and G_{1} are called measure equivalent if there exist commuting measure-preserving free actions of G_{0} and G_{1} on some infinite Lebesgue measure space (Ω, m) such that the actions of both G_{0} and G_{1} admit finite measure fundamental domains.

Remark 5.28 The notion of measure equivalence can be viewed as the measure theoretic analog of the metric notion of quasiisometric 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 compact fundamental domain Remark 5.29 There are groups which are quasi-isometric but not measure equivalent, and vice versa. Put G_{n} = (F_{3} × F_{3}) ∗ F_{n}. Then G_{m} and G_{n} are quasiisometric for n ≥ 2. Since b_{1}^{(2)} (G_{n}) = n and b_{2}^{(2)} (G_{n}) = 4, they are measure equivalent if and only if m = n. The groups ℤ and ℤ^{2} have different growth rate and are hence not quasi-isometric. Since both are infinite amenable, they are measure equivalent by a result of Ornstein and Weiss.

Theorem 5.30 (L^{2}-Betti numbers and measure equivalence, Gaboriau) Let G_{0} and G_{1} be two countable groups which are measure equivalent. Then there is a constant C > 0 such that for all p ≥ 0

b_{p}^{(2)} (G_{0}) = C · b_{p}^{(2)} (G_{1}).

Remark 5.31 The corresponding result of Gaboriau for quasi-isometry instead of measure equivalence is false. But the vanishing of the p-th L^{2}-Betti numbers is a quasiisometry invariant by a result of Pansu.

Remark 5.32 Since any infinite amenable group of measure equivalent to ℤ by a result of Ornstein and Weiss and the L^{2}- Betti numbers of ℤ vanish, we rediscover the result that the L^{2}-Betti numbers of an infinite amenable group vanish.

Conjecture 5.33 Measure equivalence and L^{2}-torsion) Let G_{0} and G_{1} be two countable groups which are measure equivalent. Suppose that there are finite models for BG_{0} and BG_{1} and b_{p}^{(2)} (G_{0}) = b_{p}^{(2)} (G_{1}) = 0 for p ≥ 0. Then

ρ^{(2)} (EG_{0}) = 0 ⇔ ρ^{(2)} (EG_{1}) = 0.

Problem 5.34 Let F and F′ be two finitely generated free groups of rank ≥ 2. Are the associated group von Neumann algebras isomorphic if and only if the groups F and F′ are isomorphic?

Definition 5.35 (Connes-Shlyakhtenko) Let A be a finite von Neumann algebra. Define its L^{2}-Betti number by

b_{p}^{(2)} (A) = dim_{A\overline{⊗}A^{op}} (HH_{p}(A; A\overline{⊗}A^{op})) .

Problem 5.36 Let G be a group. Does then the following hold

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

Of course a positive answer to Problem 5.36 would imply a positive answer to Problem 5.34.

This definition is motivated by a certain dictionary due to Connes, the following equalities

ℂG ⊗ (ℂG)^{op} = ℂ[G × G];
N (G)\overline{⊗}N (G)^{op} = N (G × G).

and the calculation

b_{p}^{(2)} (G)

= dim_{N (G)} (Tor _{p}^{ℂG}(ℂ; N (G))
= dim_{N (G)\overline{⊗}N (G)^{op}}

(Tor _{p}^{ℂG}(ℂ; N (G)) ⊗_{N (G)} N (G)\overline{⊗}N (G)^{op}

= dim_{N (G)\overline{⊗}N (G)^{op}} (Tor _{p}^{ℂG}(ℂ; N (G)\overline{⊗}N (G)^{op}))
= dim_{N (G)\overline{⊗}N (G)^{op}}

(Tor _{p}^{ℂG⊗(ℂG)^{op}} (ℂG; N (G)\overline{⊗}N (G)^{op}))

= dim_{N (G)\overline{⊗}N (G)^{op}}

(HH_{p}(ℂG; N (G)\overline{⊗}N (G)^{op})) .