Dynamics on the PSL(2, $\mathbb{C}$)-character variety of a twisted $I$-bundle

Abstract

Let $M$ be a twisted interval bundle over a closed nonorientable hyperbolizable surface. Let $\mathcal{X}(M)$ be the PSL(2, $\mathbb{C}$)-character variety of ${\pi }_1(M)$. We examine the dynamics of the action of Out$({\pi }_1(M))$ on $\mathcal{X}(M),$ 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 $M$. Furthermore, we identify which discrete and faithful representations can lie in a domain of discontinuity for the action of Out$({\pi }_1(M))$ on $\mathcal{X}(M)$.