On systems of generators of arithmetic subgroups of higher rank groups
type: Item
instance of: scholarly article
title: On systems of generators of arithmetic subgroups of higher rank groups
zbMATH Open document ID: 0822.22005
published in: Pacific Journal of Mathematics
publication date: 1995-10-05T00:00:00Z
review text: We show that if \(\Gamma\) is a noncocompact irreducible discrete subgroup with finite covolume, of a real semisimple group \(G\) of real rank at least two and \(N_ 1\) and \(N_ 2\) are two maximal unipotent subgroups of \(\Gamma\) such that \(N_ 1 \cap N_ 2 = \{1\}\), then the subgroup generated by \(N_ 1\) and \(N_ 2\) inside \(\Gamma\) has finite index in \(\Gamma\). The proof consists in constructing an extension of the congruence completion of \(G(\mathbb{Q})\) (where \(\Gamma\) is commensurate to \(G(\mathbb{Z})\)) whose open subgroups are groups generated by \(N_ 1\) and \(N_ 2\) (for various \(\Gamma \subset G(\mathbb{Z})\)). We then show that this extension is central and splits over \(G(\mathbb{Q})\). Then, by using well known results on central extensions of \(G( \widehat {\mathbb{Z}})\) we show that the extension is finite which then proves the result quoted above.
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 20G20
Mathematics Subject Classification ID: 20H05
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 759238
zbMATH Keywords: finite index subgroup
zbMATH Keywords: noncocompact irreducible discrete subgroup
zbMATH Keywords: real semisimple group
zbMATH Keywords: maximal unipotent subgroups
zbMATH Keywords: congruence completion
zbMATH Keywords: central extensions
author: T. N. Venkataramana
MaRDI profile type: Publication
full work available at URL: pjm.1994.166.193
OpenAlex ID: W2034152287
DOI: 10.2140/PJM.1994.166.193
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description: scientific article; zbMATH DE number 759238