Invariable Generation of Infinite Groups

TL;DR

This paper investigates the conditions under which infinite groups, especially linear groups, are invariably generated by finite sets, revealing a key link to virtual solvability and properties of arithmetic groups.

Contribution

It establishes that finitely generated linear groups are finitely invariably generated if and only if they are virtually solvable, and explores invariable generation in the context of arithmetic groups.

Findings

Finitely generated linear groups are invariably generated by finite sets iff virtually solvable.

Profinite completions of certain arithmetic groups are finitely invariably generated.

Provides criteria linking invariable generation to group structural properties.

Abstract

A subset S of a group G invariably generates G if G = <s^(g(s)) | s in S> for each choice of g(s) in G, s in 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.