Finitely generated profinitely dense free groups in higher rank semi-simple groups
type: Item
instance of: scholarly article
title: Finitely generated profinitely dense free groups in higher rank semi-simple groups
zbMATH Open document ID: 0940.22009
DOI: 10.1007/BF01237181
published in: Transformation Groups
publication date: 2000-07-17T00:00:00Z
review text: Let \(G\) be a non-compact linear semi-simple group and \(\Gamma\) an arithmetic subgroup. If the associated simply connected algebraic group over \({\mathbb Q}\) for \(\Gamma\) has the congruence subgroup property, the authors prove \(\Gamma\) contains a finitely generated profinitely dense free (\textit{fgpdf}) subgroup. Here profinitely dense means that the subgroup maps onto every finite quotient of \(\Gamma\). The congruence subgroup property is defined in the paper, but holds for most groups not of type \(A_n\). This work answers a question posed by Shalev regarding previous work of Margulis and Soifer. It also shows the existence of \textit{fgpdf} subgroup of \(SL_n({\mathbb Z})\) for \(n\geq 3\). More generally, they prove that \(SL_n({\mathbb Z})\) may be replaced by any irreducible arithmetic non-compact lattice in a higher rank group.
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 22E40
zbMATH DE Number: 1435999
zbMATH Keywords: semi-simple group
zbMATH Keywords: congruence subgroup property
zbMATH Keywords: profinitely dense
zbMATH Keywords: irreducible arithmetic non-compact lattice
author: G. A. Soifer
author: T. N. Venkataramana
reviewed by: Mark R. Sepanski
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1435999