The dynamics of Aut(F_n) on redundant representations

Abstract

We study some dynamical properties of the canonical Aut(F_n)-action on the space 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=SL(2,R) or SL(2,C) we show that the action is not weak mixing, in the sense that the diagonal action on R_n(G)^2 is not ergodic.

Some of the statements and arguments rely on the assumption that the algebraic group G is simply connected. This assumption, which was missing in the previous version, is not necessary in the archimedean cases, but it is needed in the non-archimedean cases