We prove the existence of positive lower bounds on the Cheeger constants of manifolds of the form where is a contractible Riemannian manifold and is a discrete subgroup, typically with infinite covolume. The existence depends on the -Betti numbers of , its subgroups, and a uniform lattice of . As an application, we show the existence of a uniform positive lower bound on the Cheeger constant of any manifold of the form where is real hyperbolic -space and is discrete and isomorphic to a subgroup of the fundamental group of a complete finite-volume hyperbolic -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 when is geometrically finite. Another application shows the existence of a uniform positive lower bound on the zeroth eigenvalue of the Laplacian of over all discrete free groups whenever is even. (The bound depends on .) This extends results of Phillips–Sarnak and Doyle, who obtained such bounds for when is a finitely generated Schottky group.
"Cheeger constants and -Betti numbers." Duke Math. J. 164 (3) 569 - 615, 15 February 2015. https://doi.org/10.1215/00127094-2871415