Finiteness theorems for discrete subgroups of bounded covolume in semi- simple groups
Let G be an absolutely almost simple simply connected k-group, and \(G'\) a k-group centrally k-isogenous to G. The authors prove the following theorem: Given any \(c>0\), there are only finitely many choices of number field k, of \(G'/k\) of absolute rank \(\geq 2\), of a finite set S of places of k containing all the archimedean places, and of arithmetic groups \(\Gamma '\subset G_ S'\) (up to conjugacy), such that \(\mu_ S'(G_ S'/\Gamma ')\leq c\). (Here \(\mu_ S'\) is a certain universal normalization of Haar measures for \(G_ S'\) proposed by J. Tits). The proof also yields analogous finiteness results in the function field case, under some mild restrictions. Motivation for this work apparently comes from work of \textit{W. M. Kantor}, \textit{R. A. Liebler} and \textit{J. Tits} [Bull. Am. Math. Soc., New. Ser. 16, 129-133 (1987; Zbl 0631.20033)] giving an explicit (finite!) list of triples (F,G,\(\Gamma\)), where F is a non-archimedean local field, G an absolutely almost simple F-group of F-rank \(\geq 2\) and \(\Gamma\) a discrete subgroup of G(F) acting transitively on the chambers of the Bruhat-Tits building of G(F). In fact, the finiteness of this list of triples (F,G,\(\Gamma\)) follows from (a special case of) the main result of the present work. A brief, technical addendum to this paper has subsequently appeared [ibid. 71, 173-177 (1990)].
scientific article; zbMATH DE number 4160841
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