Cigole Thomas, Colorado State University
Authors: Cigole Thomas
2023 AWM Research Symposium
Women in Groups, Geometry, and Dynamics [Organized by Hannah Hoganson and Rylee Lyman]

If G is a reductive algebraic group over Z, the G-character variety of a finitely presented group, F parameterizes the set of closed conjugation orbits in Hom(F, G). In this talk, we explore the dynamics of the action of the group of outer automorphisms, Out(F), on the finite field points of the character variety X_F(G) . We define groups of free-type as groups with elementary automorphisms similar to the Nielsen transformations of a free group. We then prove that the Aut(F)-action is transitive on the set of epimorphisms from F to G when F is of free-type and a condition on the number of minimal generators of F is satisfied. Finally, we introduce the idea of asymptotic ratio of an orbit to define the ratio of the number of points in the orbit to that in the variety as the order of the finite field goes to infinity. If the asymptotic ratio of an orbit equals one, we say that the action is asymptotically transitive. We conclude by providing an upper bound for the asymptotic ratio in these cases and thus prove that the action is not asymptotically transitive on the SL_n-character varieties of Z^r for n = 2, 3.

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