Dynamics of the mapping class group on the moduli of a punctured sphere with rational holonomy

Abstract

Let $M$ be a four-holed sphere and $\Gamma$ the mapping class group of $M$ fixing the boundary $\partial M$. The group $\Gamma$ acts on $M_B(SL(2,C)) = Hom_B^+(pi_1(M),SL(2,C))/SL(2,C)$ which is the space of completely reducible $SL(2,C)$-gauge equivalence classes of flat $SL(2,C)$-connections on $M$ with fixed holonomy $B$ on $\partial M$. Let $B \in (-2,2)^4$ and $M_B$ be the compact component of the real points of $M_B(SL(2,C))$. These points correspond to SU(2)-representations or $SL(2,R)$-representations. The $\Gamma$-action preserves $M_B$ and we study the topological dynamics of the $\Gamma$-action on $M_B$ and show that for a dense set of holonomy $B \in (-2,2)^4$, the $\Gamma$-orbits are dense in $M_B$. We also produce a class of representations $\rho \in \Hom_B^+(pi_1(M),SL(2,R))$ such that the $\Gamma$-orbit of $[\rho]$ is finite in the compact component of $M_B(SL(2,R))$, but $\rho(\pi_1(M))$ is dense in $SL(2,R)$.