# 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.

## Key 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.