The dynamics of Aut(F_n) on redundant representations

TL;DR

This paper investigates the dynamical properties of the automorphism group of free groups acting on a space of representations into algebraic groups over local fields, confirming minimality and ergodicity but not weak mixing in certain cases.

Contribution

It proves that the Aut(F_n) action on the space of redundant representations is always minimal and ergodic, and clarifies the lack of weak mixing in classical cases.

Findings

The action is always minimal.

The action is always ergodic.

Not weak mixing for G=SL(2,R) or SL(2,C).

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.