Volume and L-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 [5] – 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 L-Betti numbers The L-Betti numbers of a closed Riemannian manifold, as introduced by Michael 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 Michael Atiyah defines the i-th L-Betti number in terms of the heat kernel on X̃ as b (2) i (X) = lim t→∞ ∫ F trC e−t∆i(x, x)dvol(x). Subsequently, simplicial and homological definitions of L-Betti numbers were developed by Dodziuk, Farber, and Lück. An important consequence of the equivalence of these definitions is the homotopy invariance of L-Betti numbers. 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(Ap) = 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)⊗ZΓ C∗(X̃) ) ∈ [0,∞] where L(Γ) is the group von Neumann algebra of Γ. Some of the most fundamental properties of L-Betti numbers are: • π1(X) finite ⇒ b i (X) = bi(X̃)/|π1(X)| • ∑ i≥0(−1)b (2) 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 i (X) = 0. • If X is a 2n-dimensional hyperbolic manifold then b i (X) > 0 if and only if i = n.