Volume and L2-Betti numbers of aspherical manifolds

We give a leisurely account of the relationship between volume and L2-Betti numbers on closed, aspherical manifolds based on the results in [4] – albeit with a different point of view. This paper grew out of a talk presented at the first colloquium of the Courant Center in Göttingen in October 2007. 1. Review of L2-Betti numbers The L2-Betti numbers of a closed Riemannian manifold, as introduced by M. Atiyah, are analytical invariants of the long-time behavior of the heat kernel of the Laplacians of forms on the universal cover. We give a very brief review of these invariants; for extensive information the reader is referred to the standard reference [3]. Let X̃ → X be the universal cover of a compact Riemannian manifold, and let F ⊂ X̃ be a π1(X )-fundamental domain. Then Atiyah defines the i -th L2-Betti number in terms of the heat kernel on X̃ as b i (X ) = lim t→∞ ∫ F trC e −t∆i (x, x)d vol (x). Subsequently, simplicial and homological definitions of L2-Betti numbers were developed by Dodziuk, Farber, and Lück. An important consequence of the equivalence of these definitions is the homotopy invariance of L2-Betti numbers. 2000 Mathematics Subject Classification . 22D20,53C20,58J22. 98 Mathematisches Institut, Courant-ColloquiumTrends in Mathematics, 2008 Lück’s definition is based on a dimension function dimA (M) for arbitrary modules M over a finite von Neumann algebra A with trace tr : A → C. For example, one has dimA (A p) = tr(p). Lück proceeds then to define b i (X ) for an arbitrary space X with Γ=π1(X ) as (1.1) b i (X ) = dimL(Γ) Hi ( L(Γ)⊗ZΓC∗(X̃ ) ) ∈ [0,∞] where L(Γ) is the group von Neumann algebra of Γ. Some of the most fundamental properties of L2-Betti numbers are: – π1(X ) finite⇒ b i (X ) = bi (X̃ )/|π1(X )| – ∑ i≥0(−1) b i (X ) =χ(X ) = ∑ i≥0(−1) bi (X ). – X̄ → X d-sheeted cover⇒ b i (X̄ ) = d ·b (2) i (X ). – If X is aspherical and π1(X ) amenable then b (2) i (X ) = 0. – If X is a 2n-dimensional hyperbolic manifold then b i (X ) > 0 if and only if i = n. 2. Theorems relating volume and L2-Betti numbers Assumption 2.1. Throughout this section, let M be an n-dimensional, closed, aspherical manifold. The inequality of Theorem 2.2 is stated by Mikhail Gromov [2]*Section 5.33 on p. 297 along with an idea(1) which he attributes to Alain Connes. We provide the first complete proof of that inequality [4]*Corollary to Theorem A. The rigorous implementation of Gromov’s idea uses tools and ideas from Damien Gaboriau’s theory of L2-Betti numbers of measured equivalence relations and spaces with groupoid actions of such. Theorem 2.2. If (M , g ) has a lower Ricci curvature bound Ricci(M , g )≥−(n−1)g , then b i (M)≤ constn vol(M , g ) for every i ≥ 0. The minimal volume of a smooth manifold N is defined as the infimum of volumes of complete metrics on N whose sectional curvature is pinched between −1 and 1. We obtain the following (1)We refer to this idea as randomization. Roman Sauer: Volume and L2-Betti numbers of aspherical manifolds 99 Corollary 2.3 (Minimal volume estimate). b i (M)≤ constn minvol(M). The following theorem [4]*Theorem B is a generalization of a well-known vanishing result of Jeff Cheeger and Mikhail Gromov. Its connection to volume becomes apparent through its corollary. Theorem 2.4. If M is covered by open, amenable sets such that every point belongs to at most n sets, then b i (M) = 0 for every i ≥ 0. Here a subset U ⊂M is called amenable if π1(U ) maps to an amenable subgroup ofπ1(M). There is also a version of this theorem for arbitrary spaces [4]*Theorem C. The following corollary is a non-trivial implication of the theorem above and work of Mikhail Gromov [1]*Section 3.4 where he constructs amenable coverings in the presence of small volume. Corollary 2.5. There is a constant εn > 0 only depending on n such that minvol(M) < εn ⇒ b i (M) = 0 for every i ≥ 0. The results above are analogs of well-known theorems by Mikhael Gromov where L2-Betti numbers are replaced by simplicial volume. Note however that the assumption of asphericity is crucial here unlike in the case of the simplicial volume. 3. Idea of proof of the main theorem We describe some ideas involved in the proof of Theorems 2.2 and 2.4. In Subsection 3.1 we describe a general technique of bounding L2-Betti numbers by constructing suitable equivariant coverings on the universal cover. Since the assumptions of our theorems are too weak to garantuee the existence of such covers we need substantially modify this technique; the new tool runs under the name randomization, and it is explained in Subsection 3.2. A full proof based on randomization is rather long and complicated; we explain instead an instructive toy example in Subsection 3.4. A crucial property of L2-Betti numbers is described in Subsection 3.3. We conclude this sketch of proof in Subsection 3.5 with some remarks about other ingredients. Throughout the section, we refer to Assumption 2.1. 100 Mathematisches Institut, Courant-ColloquiumTrends in Mathematics, 2008 3.1. How to bound L2-Betti numbers by equivariant coverings in general. Let Γ = π1(M). Suppose we construct, under a certain geometrical assumption, a Γequivariant open covering U of the universal cover M̃ . Let us say that U = {Ui }i∈I is indexed by a free Γ-set set I , and we have γUi = Uγi . By a standard argument (partition of unity) one obtains a Γ-equivariant map f from M̃ to the nerve of U . The nerve is embedded in the full simplicial complex with index set I which we denote by ∆(I ). Let Ω= map ( M̃ ,∆(I ) ) be the space of continuous maps with the natural Γ-action. We may view f as an element in ΩΓ, the subspace ofΩ consisting of Γ-equivariant maps. Next we argue that both the i -th Betti number and the L2-Betti number are bounded from above by the number of equivariant i -simplices hit by f (M̃). Let Fi be a set of Γ-representatives of the i -skeleton ∆(I )(i ). For any g ∈Ω, let Ci (g ) ∈ N be the number of i -simplices in Fi hit by f (M̃). We think of Ci as a function Ci :Ω→Z. Since M̃ is contractible, M is a model of the classifying space BΓ, and the universal property of EΓ, the universal cover of BΓ, implies that there is an equivariant homotopy retract M̃ f !! ∆(I ) "" . Using the fact that the i -th L2-Betti number is bounded by the number of equivariant i -simplices and the fact that the L2-Betti number is some sort of dimension (with nice properties) of a certain homology module (see (1.1)), we easily obtain that b i (M)≤Ci ( f ). By going to Γ-quotients we also obtain the same estimate for the usual Betti numbers. By Poincare duality it is actually enough to control Cn( f ), and we have (3.1) bi (M),b (2) i (M)≤ constn Cn( f ) for a constant constn only depending on n. This follows from [3]*Example 14.28 on p. 498 since the fundamental class of M can be written as a sum of at most Cn( f ) singular simplices. So to get a good bound on b i (M), we should find an equivariant cover U such that for the resulting map f to the nerve the quantity Cn( f ) is rather small. Roman Sauer: Volume and L2-Betti numbers of aspherical manifolds 101 3.2. Randomization. One directly sees the limitations of the above technique. The trivial estimate Cn( f )≥ 1 for any map f ∈Ω prevents us from proving the vanishing of the L2-Betti numbers. In particular, we cannot hope to prove Theorems 2.2 and 2.4 using it. Next we phrase an idea of Mikhail Gromov (attributed to Alain Connes) in probabilistic terms that modifies the above technique. By changing the point of view a bit, we regard a map f ∈ ΩΓ that we sought to construct before as a Γ-invariant point measure on the Borel space Ω. Instead of trying to find a point measure f with small Cn( f ), Gromov suggests to look for Γinvariant probability measures μ onΩ such that the expected value