Invariable generation of prosoluble groups

Abstract

A group $G$ is invariably generated by a subset $S$ of $G$ if $G= s^{g(s)} \mid s\in S$ for each choice of $g(s) \in G$, $s \in S$. Answering two questions posed by Kantor, Lubotzky and Shalev, we prove that the free prosoluble group of rank $d \ge 2$ cannot be invariably generated by a finite set of elements, while the free solvable profinite group of rank $d$ and derived length $l$ is invariably generated by precisely $l(d-1)+1$ elements.