Zariski dense subgroups of arithmetic groups
type: Item
instance of: scholarly article
title: Zariski dense subgroups of arithmetic groups
zbMATH Open document ID: 0679.20040
DOI: 10.1016/0021-8693(87)90106-2
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: The author, motivated by \textit{G. A. Margulis} and \textit{G. A. Sojfer} [ibid. 69, 1-23, (1981; Zbl 0457.20046)], shows the existence of Zariski- dense subgroups in arithmetic groups which virtually contain some unipotent subgroups and are of infinite index. On the other hand, he proves that for some other unipotent subgroups, every Zariski-dense subgroup containing them must be of finite index. The last type of results (which are too technical to quote here) are very interesting as they give a sufficient condition for some elements to generate a finite index subgroup. For example, the author deduces that if \(n\geq 3\) and \(SL_ 3\subseteq SL_ n\) is the standard inclusion, then any Zariski- dense subgroup of \(SL_ n({\mathbb{Z}})\) which virtually contains \(SL_ 3({\mathbb{Z}})\) is of finite index in \(SL_ n({\mathbb{Z}})\).
Mathematics Subject Classification ID: 20G30
Mathematics Subject Classification ID: 20H05
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 22E40
zbMATH DE Number: 4112886
zbMATH Keywords: Zariski-dense subgroups
zbMATH Keywords: arithmetic groups
zbMATH Keywords: unipotent subgroups
zbMATH Keywords: infinite index
zbMATH Keywords: finite index subgroup
author: T. N. Venkataramana
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90106-2
OpenAlex ID: W2017708288
cites work: Finiteness of quotient groups of discrete subgroups
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cites work: Free subgroups in linear groups
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cites work: Strong approximation for Zariski-dense subgroups of semi-simple algebraic groups
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description: scientific article; zbMATH DE number 4112886