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Moduli spaces of hyperbolic 3-manifolds

We study the topology of the moduli space AI(M) of unmarked hyperbolic 3-manifolds homotopy equivalent to a fixed compact hyperbolizable 3-manifold M . This moduli space is the quotient of the more commonly studied space AH(M) of marked hyperbolic 3-manifolds homotopy equivalent to M by the action of Out(π1(M)). The deformation space AH(M) is contained in the character varietyX(M) associated to M and we also study the dynamics of the action of Out(π1(M)) on both AH(M) and on X(M). For a compact hyperbolizable 3-manifoldM , the deformation spaceAH(M) of marked hyperbolic 3-manifolds homotopy equivalent to M is a familiar object of study. This deformation space sits naturally inside the character variety X(M) and the outer automorphism group Out(π1(M)) acts by homeomorphisms on both AH(M) and X(M). The action of Out(π1(M)) on AH(M) and X(M) has largely been studied in the case when M is an interval bundle over a closed surface (see, for example, [8, 21, 45, 17]) or a handlebody (see, for example, [41, 50]). In this paper, we initiate a study of this action and its quotient for general hyperbolizable 3-manifolds. We begin by studying the topological quotient AI(M) = AH(M)/Out(π1(M)) which we may think of as the moduli space of unmarked hyperbolic 3manifolds homotopy equivalent to M . The space AH(M) is a rather pathological topological object itself, often failing to even be locally connected (see Bromberg [11] and Magid [33]). However, since AH(M) is a closed subset of an open submanifold of the character variety, it does retain many nice topological properties. We will see that the topology of AI(M) reflects the topology of M . The first hint that AI(M) may be even worse topologically than AH(M), is Thurston’s proof [47] that if M is homeomorphic to S × I, then there are infinite order elements of Out(π1(M)) which have fixed points in AH(M). (These elements are pseudo-Anosov mapping classes in Mod+(S) ⊂ Out(π1(S).) Date: November 9, 2009. The first author was partially supported by National Science Foundation grants DMS0504791 and DMS-0554239. The second author is partially supported by National Science Foundation grant DMS-0904355.