#### Volume 10, issue 2 (2006)

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

### Juan Souto and Peter Storm

Geometry & Topology 10 (2006) 715–736
 arXiv: math.GT/0504474
##### Abstract

We study the action of the mapping class group $Mod\left(S\right)$ on the boundary $\partial \mathsc{Q}$ of quasifuchsian space $\mathsc{Q}$. Among other results, $Mod\left(S\right)$ is shown to be topologically transitive on the subset $\mathsc{C}\subset \partial \mathsc{Q}$ of manifolds without a conformally compact end. We also prove that any open subset of the character variety $\mathsc{X}\left({\pi }_{1}\left(S\right),{SL}_{2}ℂ\right)$ intersecting $\partial \mathsc{Q}$ does not admit a nonconstant $Mod\left(S\right)$–invariant meromorphic function. This is related to a question of Goldman.

##### Keywords
hyperbolic geometry, mapping class group
Primary: 57M50
Secondary: 58D27