On the number of finite subgroups of a lattice
type: Item
instance of: scholarly article
title: On the number of finite subgroups of a lattice
zbMATH Open document ID: 1307.22008
DOI: 10.4171/CMH/333
author: Iddo Samet
published in: Commentarii Mathematici Helvetici
publication date: 2014-10-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1209.2484
reviewed by: Hatem Hamrouni
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 20E45
Mathematics Subject Classification ID: 57R18
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 6361419
zbMATH Keywords: Lattices in Lie groups
zbMATH Keywords: covolumes of lattices
zbMATH Keywords: Kazdan's property \((T)\)
zbMATH Keywords: irreducible lattices
zbMATH Keywords: semisimple Lie groups, conjugacy classes
zbMATH Keywords: orbifolds of non-positive curvature
zbMATH Keywords: counting problems
MaRDI profile type: Publication
arXiv ID: 1209.2484
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OpenAlex ID: W2963569901
review text: 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.
description: scientific article; zbMATH DE number 6361419