## Volumes of S-arithmetic quotients of semi-simple groups. With an appendix by Moshe Jarden and Gopal Prasad.(English)Zbl 0695.22005

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.
Reviewer: J.G.M.Mars

### MSC:

 22E46 Semisimple Lie groups and their representations 20G30 Linear algebraic groups over global fields and their integers 22E40 Discrete subgroups of Lie groups 11R23 Iwasawa theory
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