Dynamics on the $\mathrm{PSL}(2,\mathbb{C})$–character variety of a compression body

Abstract

Let $M$ be a nontrivial compression body without toroidal boundary components. Let $\mathcal{X}\left(M\right)$ be the $PSL\left(2,ℂ\right)$–character variety of ${\pi }_1\left(M\right)$. We examine the dynamics of the action of $Out\left({\pi }_1\left(M\right)\right)$ on $\mathcal{X}\left(M\right)$, and in particular, we find an open set, on which the action is properly discontinuous, that is strictly larger than the interior of the deformation space of marked hyperbolic $3$–manifolds homotopy equivalent to $M$.