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.
Bibliographical noteFunding 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.
© 2014 Elsevier Inc.
- Congruence subgroup property
- Invariable generation
- Linear groups
- Profinite groups