Cheeger constants and L2-Betti numbers

Abstract

We prove the existence of positive lower bounds on the Cheeger constants of manifolds of the form $X/\Gamma$ where $X$ is a contractible Riemannian manifold and $\Gamma <Isom\left(X\right)$ is a discrete subgroup, typically with infinite covolume. The existence depends on the $L^2$-Betti numbers of $\Gamma$, its subgroups, and a uniform lattice of $Isom\left(X\right)$. As an application, we show the existence of a uniform positive lower bound on the Cheeger constant of any manifold of the form ${\mathbb{H}}^4/\Gamma$ where ${\mathbb{H}}^4$ is real hyperbolic $4$-space and $\Gamma <Isom\left({\mathbb{H}}^4\right)$ 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 $\Gamma$ when $\Gamma$ is geometrically finite. Another application shows the existence of a uniform positive lower bound on the zeroth eigenvalue of the Laplacian of ${\mathbb{H}}^n/\Gamma$ over all discrete free groups $\Gamma <Isom\left({\mathbb{H}}^n\right)$ whenever $n\ge 4$ is even. (The bound depends on $n$.) This extends results of Phillips–Sarnak and Doyle, who obtained such bounds for $n\ge 3$ when $\Gamma$ is a finitely generated Schottky group.