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

TL;DR

This paper investigates the dynamics of the outer automorphism group action on the space of hyperbolic 3-manifolds and their character varieties, revealing how topology influences these dynamics.

Contribution

It analyzes the relationship between the topology of 3-manifolds and the dynamics of group actions on associated moduli and character spaces.

Findings

The dynamics of Out(((M))) reflect the topology of M.

The quotient space AI(M) serves as a moduli space of unmarked hyperbolic 3-manifolds.

The nature of the action varies with the topology of the underlying manifold.

Abstract

The space AH(M) of marked hyperbolic 3-manifold homotopy equivalent to a compact 3-manifold with boundary M sits inside the PSL_2(C)-character variety X(M) of \pi_1(M). We study the dynamics of the action of 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)/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.