# Commentarii Mathematici Helvetici

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**Volume 88, Issue 1, 2013, pp. 221–251**

**DOI: 10.4171/CMH/284**

Published online: 2013-01-07

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

Richard D. Canary^{[1]}and Peter A. Storm

^{[2]}(1) University of Michigan, Ann Arbor, USA

(2) New York, USA

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.

*Keywords: *Hyperbolic 3-manifolds, outer automorphism group, character variety, moduli spaces

Canary Richard, Storm Peter: Moduli spaces of hyperbolic 3-manifolds and dynamics on character varieties. *Comment. Math. Helv.* 88 (2013), 221-251. doi: 10.4171/CMH/284