Commensurability classes of discrete arithmetic hyperbolic groups

In this paper we show that the commensurability classes of discrete arithmetic subgroups of Isom(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.