Ergodicity conjecture for outer automorphisms of free groups

From papers

Let π\pi be a free group of rank rr, let GG be a compact connected Lie group, and let Hom(π,G)\mathsf{Hom}(\pi,G) carry the probability measure induced by Haar measure on GG.

Free-group ergodicity conjecture. If r3r\geq 3, the action of Out(π)\mathsf{Out}(\pi) on Hom(π,G)\mathsf{Hom}(\pi,G) is ergodic.

This is the free-group analogue of the ergodicity phenomenon for mapping class groups acting on surface-group representation spaces. The source gives no resolution of the claim.

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Sources & referencesView supporting material

Primary source

William M. Goldman, “Mapping Class Group Dynamics on Surface Group Representations”, arXiv:math/0509114 (2006).

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