#### Volume 14, issue 4 (2014)

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

### Michelle Lee

Algebraic & Geometric Topology 14 (2014) 2149–2179
##### Abstract

Let $M$ be a nontrivial compression body without toroidal boundary components. Let $\mathsc{X}\left(M\right)$ 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 $\mathsc{X}\left(M\right)$, 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 $M$.

##### Keywords
compression body, hyperbolic $3$–manifold, character variety, outer automorphism group
Primary: 57M50
Secondary: 57M60