1.2. TH E MAIN THEOREM S FOR HAKEN 3-MANIFOLDS 3 By GF(7Ti(M), 7Ti(P)) we denote the set of geometrically finite uniformizations of manifold pairs homotopy equivalent to (M, P). The complete forms of our mo- tivating hyperbolic questions are the following: Hyperbolic Questions: For which pared manifolds (M, P) does GF(M, P) have finitely many components? For which (M,P) does GF(7TI(M),7TI(P)) have finitely many components? A manifold pair (M,P) is a specific case of Johannson's general theory of manifolds with boundary patterns. A boundary pattern ^ on M is a collection of submanifolds of dM such that any two elements intersect in a (possibly empty) collection of arcs and circles, while any three elements intersect in a finite collection of points. One can define Out(7Ti(M),7Ti(^)) and 7£(M, m) much as above. One then asks the general topological question: Finite Index Realization Problem: For which compact orientable irreducible 3-manifolds with boundary pattern (M,gl) does the subgroup lZ(M,m) realizable by homeomorphisms have finite index in Out(7Ti(M),7Ti(^)) ? We will answer this topological question for a variety of boundary patterns, which includes all pared 3-manifolds. Full statements of the results are given in chapter 8. 1.2. The main theorems for Haken 3-manifolds In this section we will develop the notation necessary to give a complete state- ment of our results for Haken 3-manifolds with nonempty boundary but empty "boundary pattern". We will also provide outlines of their proofs, which serve as outlines for the proofs of the more general forms of our results. We first introduce more formally the notation involved in our hyperbolic ques- tion. Let M be a compact, oriented, hyperbolizable 3-manifold. Let V(TTI(M)) denote the space of discrete, faithful representations of ni(M) into PSL(2,C) and let AH(TTI(M)) denote P(7Ti(M))/PSL(2,C) where PSL(2,C) acts by conjugation. If p G AH(TTI(M)), then let Np denote B3 / p(ni(M)). Further denote by tt(p) the maximal open subset of C on which p(ni(M)) acts discontinuously, and let Np = (M3U^(p))/p(7Ti(M)). We call Np the conformal extension of Np. When Np is compact, p is said to be convex cocompact. Let CC(TTI(M)) denote the space of convex cocompact elements of AH(7Ti(M)). We further define CC(M) to consist of all p G CC(7Ti(M)) such that there exists an orientation-preserving homeomorphism from M to Np. We also consider the space A(M) of oriented compact irreducible 3-manifolds homotopy equivalent to M. Two elements M\ and M2 of A(M) are regarded as equivalent if there exists an orientation-preserving homeomorphism from M\ to M2. Now we will define the space A(M) of marked, oriented, compact, irreducible 3-manifolds homotopy equivalent to M. Its basic objects are pairs (M1\h') where M' G A(M) and h'\ M M' is a homotopy equivalence. Two pairs (Mi, hi) and (M2, /12) are considered equivalent when there exists an orientation-preserving homeomorphism fi: Mi M2 such that fi o hi is homotopic to /i2- Then A(M) is the set of all equivalence classes of such pairs. Since each element M' of A(M) is a K(TT, 1) every element a of Out(7Ti(M/)) is realized by a homotopy equivalence ha : M' —* M'. Let Ao(M') be the set of elements of A(M) of the form (M , h') for some homotopy equivalence h' : M M'.
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