On the number of finite subgroups of a lattice
semisimple Lie groups, conjugacy classes
The paper under review is concerned with upper bounds of the number \(f(\Gamma)\) of conjugacy classes of maximal finite subgroups of a lattice in a connected semisimple Lie group \(G\). The author proves the following results. {\parindent=2,5cm \begin{itemize}\item[Theorem] If \(G\) is a connected semisimple Lie group with finite center and without compact factors, then there exists a constant \(c\), which depends only on \(G\), such that \(f(\Gamma)\leq c ~\mathrm{vol}(G/\Gamma)\). \item[Theorem] If moreover the group \(G\) has Kazdan's property \((T)\) and \(\mathbb{R}\)-\(\mathrm{rank}(G)\geq 2\), then for every sequence \((\Gamma_n)\) of pairwise nonconjugate irreducible lattices in \(G\), we have \(\lim f(\Gamma_n) ~\mathrm{vol}(G/\Gamma)^{-1}=0\). NEWLINENEWLINE\end{itemize}} The paper contains a geometric interpretation of the above results.
scientific article; zbMATH DE number 6361419
scientific article