Dynamics of the mapping class group action on the variety of $\mathrm{PSL}_2 \mathbb{C}$ characters

Abstract

We study the action of the mapping class group $Mod\left(S\right)$ on the boundary $\partial \mathcal{Q}$ of quasifuchsian space $\mathcal{Q}$. Among other results, $Mod\left(S\right)$ is shown to be topologically transitive on the subset $\mathcal{C}\subset \partial \mathcal{Q}$ of manifolds without a conformally compact end. We also prove that any open subset of the character variety $\mathcal{X}\left({\pi }_1\left(S\right),{SL}_2ℂ\right)$ intersecting $\partial \mathcal{Q}$ does not admit a nonconstant $Mod\left(S\right)$–invariant meromorphic function. This is related to a question of Goldman.