Mathematics – Geometric Topology
Scientific paper
1998-06-13
Ann. of Math. (2) 152 (2000), no. 3, 693--741
Mathematics
Geometric Topology
49 pages, published version
Scientific paper
Let M be a compact, hyperbolizable 3-manifold with nonempty incompressible boundary and let AH(\pi_1(M)) denote the space of (conjugacy classes of) discrete faithful representations of \pi_1(M) into PSL 2 (C). The components of the interior MP(\pi_1(M)) of AH(\pi_1(M)) (as a subset of the appropriate representation variety) are enumerated by the space A(M) of marked homeomorphism types of oriented, compact, irreducible 3-manifolds homotopy equivalent to M. In this paper, we give a topological enumeration of the components of the closure of MP(\pi_1(M)) and hence a conjectural topological enumeration of the components of AH(\pi_1(M)). We do so by characterizing exactly which changes of marked homeomorphism type can occur in the algebraic limit of a sequence of isomorphic freely indecomposable Kleinian groups. We use this enumeration to exhibit manifolds M for which AH(\pi_1(M)) has infinitely many components.
Anderson James W.
Canary Richard D.
McCullough Darryl
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