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

## TL;DR

This paper computes Cheeger constants of hyperbolic surfaces linked to arithmetic groups and reflection groups, exploring their relation to small eigenvalues of Maass cusp forms through numerical analysis.

## Contribution

It provides the first systematic computation of Cheeger constants for these hyperbolic surfaces and investigates their connection to small eigenvalues of Maass cusp forms.

## Key findings

- Cheeger constants are computed for specific hyperbolic surfaces.
- Evidence suggests the existence of small eigenvalues.
- Numerical methods identify corresponding Maass cusp forms.

## Abstract

We compute the Cheeger constants of a collection of hyperbolic surfaces corresponding to maximal non-compact 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.