On the proportionality of covolumes of discrete subgroups (Q1067021)

If \(G_ S=\prod_{v\in S}G_ v\) is a Lie group without compact factors, where the finitely many \(G_ v\), \(v\in S\), are real or p-adic semisimple Lie groups, and if \(\Gamma_ S\subset G_ S\) is a discrete subgroup of finite covolume with respect to some invariant measure on \(G_ S\), then it is shown that there exist an invariant measure \(\omega\) on \(G_ S\) such that all subgroups commensurable to \(\Gamma_ S\) have integral covolume with respect to \(\omega\). For non arithmetic irreducible subgroups \(\Gamma_ S\) this is an immediate consequence of an old result of the first author. For arithmetic subgroups the proof depends on the theory of Bruhat and Tits on p-adic Lie groups. As a step of the proof a classification of maximal arithmetic subgroups is given.