A few remarks on invariable generation in infinite groups

TL;DR

This paper investigates invariable generation in infinite groups, focusing on topological groups like Lie groups and automorphism groups of trees, establishing which are topologically invariably generated and which are not.

Contribution

It demonstrates that certain topological groups such as SL(2,R) and automorphism groups of trees are topologically invariably generated, while others like PSL_m(K) are not.

Findings

SL(2,R) is topologically invariably generated

Automorphism groups of regular trees are topologically invariably generated

PSL_m(K) is not invariably generated for certain fields

Abstract

A subset $S$ of a group $G$ invariably generates $G$ if $G$ is generated by $\{ s^g(s) | s\in S\} $ for any choice of $g(s)\in G, s\in S$. In case $G$ is topological one defines similarly the notion of topological invariable generation. A topological group $G$ is said to be $\mathcal{TIG}$ if it is topologically invariably generated by some subset $S\subseteq G$. 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 $\mathrm{SL}_{2}(\mathbb{R})$ and the automorphism group of a regular tree are $\mathcal{TIG}$, and that the groups $PSL_m(K) ,m\geq 2$ are not $\mathcal{IG}$ for certain countable fields of infinite transcendence degree over the prime field.