Invariable generation of infinite groups

William M. Kantor; Alexander Lubotzky; Aner Shalev

doi:10.1016/j.jalgebra.2014.08.030

Invariable generation of infinite groups

William M. Kantor, Alexander Lubotzky^*, Aner Shalev

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

17 Scopus citations

Abstract

A subset S of a group G invariably generates G if G=〈s^g(s)|s∈S〉 for each choice of g(s)∈G, s∈S. In this paper we study invariable generation of infinite groups, with emphasis on linear groups. Our main result shows that a finitely generated linear group is invariably generated by some finite set of elements if and only if it is virtually solvable. We also show that the profinite completion of an arithmetic group having the congruence subgroup property is invariably generated by a finite set of elements.

Original language	American English
Pages (from-to)	296-310
Number of pages	15
Journal	Journal of Algebra
Volume	421
DOIs	https://doi.org/10.1016/j.jalgebra.2014.08.030
State	Published - 1 Jan 2015

Bibliographical note

Funding Information:
The authors acknowledge partial support from ERC Advanced Grants 226135 (A.L.) and 247034 (A.S., W.M.K.), and ISF grant 1117/13 (A.L. and A.S.). The first author is grateful for the warm hospitality of the Hebrew University while this paper was being written.

Publisher Copyright:
© 2014 Elsevier Inc.

Keywords

Congruence subgroup property
Invariable generation
Linear groups
Profinite groups

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10.1016/j.jalgebra.2014.08.030

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abstract = "A subset S of a group G invariably generates G if G=〈sg(s)|s∈S〉 for each choice of g(s)∈G, s∈S. In this paper we study invariable generation of infinite groups, with emphasis on linear groups. Our main result shows that a finitely generated linear group is invariably generated by some finite set of elements if and only if it is virtually solvable. We also show that the profinite completion of an arithmetic group having the congruence subgroup property is invariably generated by a finite set of elements.",

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Invariable generation of infinite groups

Abstract

Bibliographical note

Keywords

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