Volumes of S-arithmetic quotients of semi-simple groups. With an appendix by Moshe Jarden and Gopal Prasad (Q909806)

Let k be a global field and G a semi-simple algebraic group defined over k. Let S be a finite set of places of k containing the archimedean ones. For G simply connected and absolutely quasi-simple a formula is given for the volume of \(G(k_ S)/\Lambda\), where \(\Lambda\) is an S-arithmetic subgroup of G(k). The derivation of this formula is possible thanks to the theory of reductive groups over local fields of Bruhat and Tits. The same computations give a bound for class numbers.