Moduli spaces of hyperbolic 3-manifolds and dynamics on character varieties

Abstract

The space $AH (M)$ of marked hyperbolic 3-manifold homotopy equivalent to a compact 3-manifold with boundary $M$ sits inside the $\mathrm{PSL}_2({\mathbb{C}})$-character variety $X(M)$ of $\pi_1(M)$. We study the dynamics of the action of $\mathrm{Out}(\pi_1(M))$ on both $AH (M)$ and $X(M)$. The nature of the dynamics reflects the topology of $M$.

The quotient $AI (M)=AH (M)/\mathrm{Out}(\pi_1(M))$ may naturally be thought of as the moduli space of unmarked hyperbolic 3-manifolds homotopy equivalent to $M$ and its topology reflects the dynamics of the action.