Cheeger constants and L2-Betti numbers

Abstract

We prove the existence of positive lower bounds on the Cheeger constants of manifolds of the form X/Γ where X is a contractible Riemannian manifold and Γ<Isom(X) is a discrete subgroup, typically with infinite covolume. The existence depends on the L2-Betti numbers of Γ, its subgroups, and a uniform lattice of Isom(X). As an application, we show the existence of a uniform positive lower bound on the Cheeger constant of any manifold of the form H4/Γ where H4 is real hyperbolic 4-space and Γ<Isom(H4) is discrete and isomorphic to a subgroup of the fundamental group of a complete finite-volume hyperbolic 3-manifold. Via Patterson–Sullivan theory, this implies the existence of a uniform positive upper bound on the Hausdorff dimension of the conical limit set of such a Γ when Γ is geometrically finite. Another application shows the existence of a uniform positive lower bound on the zeroth eigenvalue of the Laplacian of Hn/Γ over all discrete free groups Γ<Isom(Hn) whenever n≥4 is even. (The bound depends on n.) This extends results of Phillips–Sarnak and Doyle, who obtained such bounds for n≥3 when Γ is a finitely generated Schottky group.