Algebra/Topology seminar

Speaker: Juan Souto (Universite de Rennes 1)

Title: Ergodicity of the mapping class group action on a component of the character variety

Abstract: Goldman proved that the variety $X_g$ of characters of representations of the fundamental group of a surface of genus $g$ into $PSL_2R$ has precisely $4g-3$ connected components $X_g(2-2g),...,X_g(2g-2)$ where moreover the component $X_g(k)$ consists of those representations with Euler number $k$. The two extremal component $X_g(2-2g)$ and $X_g(2g-2)$ are Teichmueller spaces and hence the mapping class group acts discretely on them. On the other hand Goldman conjectured that the action of the mapping class group on each one of the remaining components. I will prove that Goldman's conjecture holds true for the component $X_g(0)$ corresponding to representations with vanishing Euler number.