Mathematics Colloquium – Dynamics on Character Varieties over Finite Fields

Speaker: Cigole Thomas, Bates College

Abstract: Character varieties arise as spaces that parametrize group homomorphisms from a finitely presented group Γ into an “nice” algebraic group G, up to a natural notion of equivalence. In this talk, we explore these spaces by studying their points over finite fields Fq and examining how symmetries of the group Γ, encoded by its outer automorphism group Out(Γ), act on them.
We introduce a natural stratification of the space and count the number of points in each stratum, using these counts to gain insight into the asymptotic behavior (when q goes to ∞) of the action as the size of the field grows. In particular, we consider the case where Γ = Zr and G = SL3(C).

The talk will be self-contained and aimed at a broad audience, with the necessary background developed along the way.