A few remarks on invariable generation in infinite groups

Abstract

A subset S of a group G invariably generates G if G is generated by {sg(s)|s S} for any choice of g(s) G,s S. A topological group G is said to be ϵ if it is invariably generated by some subset S = G, and ϵ if it is topologically invariably generated by some subset S = G. 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 SL2() and the automorphism group of a regular tree are ϵ, and that the groups PSLm(K),m ≥ 2 are not ϵ for countable fields of infinite transcendence degree over a prime field.