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 $Γ$ the mapping class group of $M$ fixing the boundary $\partial M$. The group $Γ$ 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 $Γ$-action preserves $M_B$ and we study the topological dynamics of the $Γ$-action on $M_B$ and show that for a dense set of holonomy $B \in (-2,2)^4$, the $Γ$-orbits are dense in $M_B$. We also produce a class of representations $ρ\in \Hom_B^+(pi_1(M),SL(2,R))$ such that the $Γ$-orbit of $[ρ]$ is finite in the compact component of $M_B(SL(2,R))$, but $ρ(π_1(M))$ is dense in $SL(2,R)$.