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On the topology of deformation spaces of Kleinian groups

On the topology of deformation spaces of Kleinian groups
On the topology of deformation spaces of Kleinian groups
Let M be a compact, hyperbolizable 3-manifold with nonempty incompressible boundary and fundamental group G, and let AH(G) denote the space of (conjugacy classes of) discrete faithful representations of G into PSL(2, C). The components of the interior MP(G) of AH(G) (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(G) and hence a conjectural topological enumeration of the components of AH(G). 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(G) has infinitely many components.
693-741
Anderson, James W.
739c0e33-ef61-4502-a675-575d08ee1a98
Canary, Richard D.
a26b9c05-b5d3-4837-a48e-0e7bb788c0b9
McCullough, Darryl
ed3c39c0-9be3-4f46-b5d2-7b23a2e0bb53
Anderson, James W.
739c0e33-ef61-4502-a675-575d08ee1a98
Canary, Richard D.
a26b9c05-b5d3-4837-a48e-0e7bb788c0b9
McCullough, Darryl
ed3c39c0-9be3-4f46-b5d2-7b23a2e0bb53

Anderson, James W., Canary, Richard D. and McCullough, Darryl (2000) On the topology of deformation spaces of Kleinian groups. Annals of Mathematics, 152 (3), 693-741.

Record type: Article

Abstract

Let M be a compact, hyperbolizable 3-manifold with nonempty incompressible boundary and fundamental group G, and let AH(G) denote the space of (conjugacy classes of) discrete faithful representations of G into PSL(2, C). The components of the interior MP(G) of AH(G) (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(G) and hence a conjectural topological enumeration of the components of AH(G). 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(G) has infinitely many components.

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Published date: 2000

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Local EPrints ID: 29872
URI: https://eprints.soton.ac.uk/id/eprint/29872
PURE UUID: fcc5b5a2-ebb4-4c17-83c7-549bdee6fa70
ORCID for James W. Anderson: ORCID iD orcid.org/0000-0002-7849-144X

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Date deposited: 20 Jul 2006
Last modified: 17 Jul 2018 00:35

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Contributors

Author: Richard D. Canary
Author: Darryl McCullough

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