Ergodicity and conullity conjectures for redundant representations

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Let FnF_n be a free group, let GG be the group under consideration, and let X(G)X(G) be the corresponding representation space. Let RR be the set of redundant representations, namely representations ρ:FnG\rho:F_n\to G for which some proper free factor A<FnA<F_n has dense image ρ(A)\rho(A) in GG. Let PS\mathcal{P}\mathcal{S} denote the primitive-stable representations.

Redundant-representation conjecture. (a) The action of Out(Fn)\operatorname{Out}(F_n) on RR is ergodic.

(b) RPSR\cup\mathcal{P}\mathcal{S} is conull in X(G)X(G).

These assertions are motivated by the contrast between proper discontinuity on Schottky and primitive-stable representations and the expected ergodic behavior on redundant representations. The excerpt does not give a resolution, so both assertions remain open.

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Sources & referencesView supporting material

Primary source

Alexander Lubotzky, “Dynamics of Aut(Fn) Actions on Group Presentations and Representations”, arXiv:1109.0155 (2011).

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