Dynamics on nilpotent character varieties

TL;DR

This paper studies the dynamics of the outer automorphism group on the moduli space of representations of finitely generated nilpotent groups into compact Lie groups, revealing conditions for mixing behavior.

Contribution

It establishes the existence of a natural Out(N)-invariant measure on the moduli space and proves mixing of the Out(N) action when hyperbolic elements are present.

Findings

Existence of a natural Out(N)-invariant measure on the moduli space.

Out(N) action is mixing under certain conditions.

Conditions for hyperbolic elements in Out(N).

Abstract

Let R(N,G) be the connected component of the identity of the variety of representations of a finitely generated nilpotent group N into a connected compact Lie group G, and let X(N,G) be the corresponding moduli space. We show that there exists a natural Out(N)-invariant measure on X(N,G) and that whenever Out(N) has at least one hyperbolic element, the action of Out(N) on X(N,G) is mixing with respect to this measure.