Commensurability classes of discrete arithmetic hyperbolic groups

Abstract

In this paper we show that the commensurability classes of discrete arithmetic subgroups of Isom$({\mathbb{H}}^n)$ of the simplest type, i.e., those coming from quadratic forms of signature $(n,1)$ over totally real number fields, can be parametrised by isomorphisms classes of quaternion algebras. This is applied to some low dimensional examples to show the commensurability of certain Coxeter groups.