Rigidity of the dynamics of ${{\rm Aut}}({\mathsf{F}}_n)$ on representations into a compact group

TL;DR

This paper studies the action of automorphisms of free groups on the space of group homomorphisms into compact Lie groups, revealing algebraic stabilization phenomena similar to Ratner's theorems for large free group rank.

Contribution

It characterizes the orbit closures and invariant measures of the automorphism group action on representation spaces for large free group rank, showing algebraic stabilization.

Findings

Orbit closures are algebraic

Invariant measures are algebraic

Dynamics stabilize for large n

Abstract

Let $G$ be a compact Lie group. Let ${\mathsf{F}}_n$ be the free group of rank $n$. We describe the orbits of ${\mathsf{Aut}}(\mathsf{F}_n)$ on ${\mathsf{Hom}}(\mathsf{F}_n;G)$ when $n$ is sufficiently large. The dynamics stabilizes: orbit closures and invariant probability measures are algebraic, as in Ratner's theorems.