CHAPTE R 1 Introduction 1.1. Motivatio n Although most of the work that follows involves results of a purely topological nature, it was motivated by questions arising in the deformation theory of hyper- bolic 3-manifolds. We begin by introducing these questions and then proceed to the topological question which will occupy most of our attention. We will say that a compact, oriented, irreducible 3-manifold M is hyperboliz- able if its interior admits a (complete) hyperbolic structure. Thurston has shown that a compact, oriented, irreducible 3-manifold with nonempty boundary is hy- perbolizable if and only if it is atoroidal. If M has no torus boundary components then there is a convex cocompact uniformization of M , i. e. a discrete faithful rep- resentation p: 7Ti(M) —• PSL(2,C) such that there exists a orientation-preserving homeomorphism from M to (H 3 U Q(p))/P(TTI(M)) (where tt(p) is the domain of discontinuity for the action of p{ni{M)) on C). We will be interested in studying the space CC(M ) of all convex cocompact uniformizations of M. The work of Ahlfors, Bers, Kra, Mar den and Maskit allows one to give an explicit parameterization of CC(M) . Each component of CC(M ) is homeomorphic to T(dM)/ Modo(M) where T(dM) is the Teichmiiller space of all hyperbolic structures on dM and Modo(M) is the group of isotopy classes of orientation-preserving homeomorphisms of M which are homotopic to the identity. Teichmiiller space is a finite-dimensional cell and Modo(M) acts discontinuously and freely on T(dM), so each component of CC(M) is a finite-dimensional manifold. If M has incompressible boundary, then Modo(M) is trivial, so T(dM) is simply a cell. The most basic form of our motivating question is as follows: Hyperboli c Questio n (convex cocompac t case): For which compact hyper- bolizable 3-manifolds M without torus boundary components does the space CC(M ) of convex cocompact uniformizations of M have finitely many components ? To give our topological enumeration of the components of CC(M ) we intro- duce the following notation. The outer automorphism group Out(7Ti(M)) is equal to the quotient of the group Aut(7Ti(M)) of automorphisms of TTI(M) by the normal subgroup Inn(7Ti(M)) of inner automorphisms of 7Ti(M). If h\ M —• M is a home- omorphism, then h determines a well-defined element of Out(7Ti(M)), although it does not determine a well-defined element of Aut(7ri(M)) unless we make a choice of basepoint for M which is preserved by h. In this case we will say that the element of Out(7Ti(M)) determined by h is realized by h. By 1Z{M) we denote the subgroup of Out(7Ti(M)) consisting of elements which are realized by homeomorphisms of M , and by + (M) the subgroup realizable by orientation-preserving homeomorphisms. 1
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