A few remarks on invariable generation in infinite groups

Abstract

A subset [Formula: see text] of a group [Formula: see text] invariably generates [Formula: see text] if [Formula: see text] is generated by [Formula: see text] for any choice of [Formula: see text]. A topological group [Formula: see text] is said to be [Formula: see text] if it is invariably generated by some subset [Formula: see text], and [Formula: see text] if it is topologically invariably generated by some subset [Formula: see text]. In this paper, we study the problem of (topological) invariable generation for linear groups and for automorphism groups of trees. Our main results show that the Lie group [Formula: see text] and the automorphism group of a regular tree are [Formula: see text], and that the groups [Formula: see text] are not [Formula: see text] for countable fields of infinite transcendence degree over a prime field.