On deformations of free groups in compact Lie groups

TL;DR

This paper investigates the deformation spaces of free groups within compact Lie groups, proving key conjectures about the density of certain points and the ergodic action of automorphism groups.

Contribution

It proves conjectures by Margulis-Soifer and Goldman regarding the density and ergodicity properties in deformation varieties of free groups in compact Lie groups.

Findings

Non-virtually free points are dense in the deformation variety.

The action of Aut(Fn) is ergodic on the variety for n>2.

Confirmed conjectures about deformation properties in compact Lie groups.

Abstract

We study some properties of the varieties of deformations of free groups in compact Lie groups. In particular we prove a conjecture of Margulis and Soifer about the density of non-virtually free points in such variety, and a conjecture of Goldman on the ergodicity of the action of Aut(Fn) on such variety when n>2.