Representations of surface groups with finite mapping class group orbits

Abstract

Let $(S,\ast )$ be a closed oriented surface with a marked point, let $G$ be a fixed group, and let $\rho :{\pi }_1(S)⟶G$ be a representation such that the orbit of $\rho$ under the action of the mapping class group $Mod(S,\ast )$ is finite. We prove that the image of $\rho$ is finite. A similar result holds if ${\pi }_1(S)$ is replaced by the free group $F_n$ on $n\ge 2$ generators and where $Mod(S,\ast )$ is replaced by $Aut(F_n)$. We thus resolve a well-known question of M. Kisin. We show that if $G$ is a linear algebraic group and if the representation variety of ${\pi }_1(S)$ is replaced by the character variety, then there are infinite image representations which are fixed by the whole mapping class group.

6 pgs no figs