A few remarks on invariable generation in infinite groups

A subset $S$ of a group $G$ invariably generates $G$ if $G$ is generated by $\{s^g(s)\mathrm{\mid }s\in S\}$ 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 $\mathcal{T}\mathcal{I}\mathcal{G}$ 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 ${\mathrm{S}\mathrm{L}}_2(\mathbb{R})$ and the automorphism group of a regular tree are $\mathcal{T}\mathcal{I}\mathcal{G}$, and that the groups $PS{L}_m(K),m\ge 2$ are not $\mathcal{I}\mathcal{G}$ for certain countable fields of infinite transcendence degree over the prime field.