The dynamics of $\mathrm{Aut}(F_n)$ on redundant representations

Abstract

We study some dynamical properties of the canonical $\mathrm{Aut}(F_n)$-action on the space ${\mathcal{R}}_n(G)$ of redundant representations of the free group $F_n$ in $G$, where $G$ is the group of rational points of a simple algebraic group over a local field. We show that this action is always minimal and ergodic, confirming a conjecture of A. Lubotzky. On the other hand for the classical cases where $G={\mathrm{SL}}_2(\mathbb{R})$ or ${\mathrm{SL}}_2(\mathbb{C})$ we show that the action is not weak mixing, in the sense that the diagonal action on ${\mathcal{R}}_n(G{)}^2$ is not ergodic.