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

### Tsachik Gelander

The Hebrew University of Jerusalem, Israel### Yair N. Minsky

Yale University, New Haven, United States

## 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.

## Cite this article

Tsachik Gelander, Yair N. Minsky, The dynamics of $\operatorname{Aut}(F_n)$ on redundant representations. Groups Geom. Dyn. 7 (2013), no. 3, pp. 557–576

DOI 10.4171/GGD/197