Invariable generation of finite classical groups

Abstract

A subset of a group invariably generates the group if it generates even when we replace the elements by any of their conjugates. In a 2016 paper, Pemantle, Peres and Rivin show that the probability that four randomly selected elements invariably generate $S_n$ is bounded away from zero by an absolute constant for all $n$. Subsequently, Eberhard, Ford and Green have shown that the probability that three randomly selected elements invariably generate $S_n$ tends to zero as $n \rightarrow \infty$. In this paper, we prove an analogous result for the finite classical groups. More precisely, let $G_r(q)$ be a finite classical group of rank $r$ over $\mathbb{F}_q$. We show that for $q$ large enough, the probability that four randomly selected elements invariably generate $G_r(q)$ is bounded away from zero by an absolute constant for all $r$, and for three elements the probability tends to zero as $q \rightarrow \infty$ and $r \rightarrow \infty$. We use the fact that most elements in $G_r(q)$ are separable and the well-known correspondence between classes of maximal tori containing separable elements in classical groups and conjugacy classes in their Weyl groups.