Cheeger constants of arithmetic hyperbolic 3-manifolds

Abstract

We study the Cheeger constants of certain infinite families of arithmetic hyperbolic three-manifolds, as well as certain graphs associated to these manifolds. We derive computable bounds on the Cheeger constants, and therefore bounds on the first eigenvalue of the Laplacian, by adapting discrete methods due to Brooks, Perry and Petersen. We then modify probabilistic methods due to Brooks and Zuk to obtain sharper, asymptotic bounds. A consequence is that the Cheeger constants are quite small, implying that Cheeger’s inequality is generally insufficient to prove Selberg’s eigenvalue conjecture.