Cheeger constants of hyperbolic reflection groups and Maass cusp forms of small eigenvalues

Abstract

We compute the Cheeger constants of a collection of hyperbolic surfaces corresponding to maximal non-cocompact arithmetic Fuchsian groups, and to subgroups which are the rotation subgroup of maximal reflection groups. The Cheeger constants are geometric quantities, but relate to the smallest eigenvalues of Maass cusp forms. From geometrical considerations, we find evidence for the existence of small eigenvalues. We search for these small eigenvalues and compute the corresponding Maass cusp forms numerically.