On the set of discrete subgroups of bounded covolume in a semisimple group
type: Item
instance of: scholarly article
title: On the set of discrete subgroups of bounded covolume in a semisimple group
zbMATH Open document ID: 0658.22004
DOI: 10.1007/BF02837812
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1987-01-01T00:00:00Z
review text: Let S be a finite set and let, for each \(s\in S\), there be given a local field \(k_ s\) of characteristic zero and an almost absolutely simple isotropic \(k_ s\)-group \({\mathfrak G}_ s\). We view \(G_ s={\mathfrak G}_ s(k_ s)\) as a locally compact topological group, using the topology of \(k_ s\), and let G be the product of the \(G_ s\). For \(T\subset S\) let \(G_ T=\prod_{s\in T}G_ s\). We say that a discrete subgroup L of G is ``irreducible'' if there is no partition \(S=A\cup B\) into non-empty subsets such that \((L\cap G_ A)(L\cap G_ B)\) is of finite index in G. We let \(r_ s\) denote the \(k_ s\)-rank of \({\mathfrak G}_ s\) and r(G) the sum of the \(r_ s\) (s\(\in S)\). The following theorem is proved in the paper. Theorem. Fix a constant \(c>0\). Assume r(G)\(\geq 2\). Then the irreducible discrete subgroups L of G with covolume c(L)\(\leq c\) form finitely many conjugacy classes. For real groups this theorem was proved earlier by H. C. Wang. As in Wang's proof, the proof of the theorem is based on the fact that L is finitely presented and locally rigid.
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 20G25
zbMATH DE Number: 4075418
zbMATH Keywords: local field
zbMATH Keywords: almost absolutely simple isotropic \(k_ s\)-group
zbMATH Keywords: locally compact topological group
zbMATH Keywords: irreducible discrete subgroups
zbMATH Keywords: covolume
zbMATH Keywords: finitely presented
zbMATH Keywords: locally rigid
MaRDI profile type: Publication
cites work: Density properties for certain subgroups of semi-simple groups without compact components
cites work: Corners and arithmetic groups (Appendice: Arrondissement des varietes a coins par A. Douady et L. Herault)
cites work: Homomorphismes ''abstraits'' de groupes algébriques simples
cites work: Q5537437
cites work: Q5655596
cites work: Presentations for groups acting on simply-connected complexes
cites work: Discrete groups of motions of manifolds of nonpositive curvature
cites work: Strong approximation for semi-simple groups over function fields
cites work: Elementary proof of a theorem of Bruhat-Tits-Rousseau and of a theorem of Tits
cites work: Q3635349
cites work: Algebraic and abstract simple groups
cites work: Q5639945
cites work: On density properties of S-subgroups of locally compact groups
cites work: Adèles and algebraic groups. (Appendix 1: The case of the group \(G_2\), by M. Demazure. Appendix 2: A short survey of subsequent research on Tamagawa numbers, by T. Ono)
cites work: Q3687381
Recommended article: Finiteness theorems for discrete subgroups of bounded covolume in semi- simple groups
Recommended article: Addendum to: Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups
Recommended article: On the proportionality of covolumes of discrete subgroups
Recommended article: The covolume of discrete subgroups of \(\mathrm{Iso}(\mathbb H^{2m})\)
Recommended article: Q3754831
Recommended article: A note on bounded-cohomological dimension of discrete groups
Recommended article: Q3789808
Recommended article: Q4832440
Recommended article: Q3331934
author: Armand Borel
description: scientific article; zbMATH DE number 4075418