Groups, Geometry, and Dynamics


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Volume 7, Issue 3, 2013, pp. 557–576
DOI: 10.4171/GGD/197

Published online: 2013-08-27

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

Tsachik Gelander[1] and Yair N. Minsky[2]

(1) The Hebrew University of Jerusalem, Israel
(2) Yale University, New Haven, United States

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.

Keywords: $\operatorname{Aut}(F_n)$, character varieties, algebraic groups, 3-dimensional topology, dynamical decomposition, redundant representation, ergodicity

Gelander Tsachik, Minsky Yair: The dynamics of $\operatorname{Aut}(F_n)$ on redundant representations. Groups Geom. Dyn. 7 (2013), 557-576. doi: 10.4171/GGD/197