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.