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Cores of hyperbolic 3-manifolds and limits of Kleinian groups II

Cores of hyperbolic 3-manifolds and limits of Kleinian groups II
Cores of hyperbolic 3-manifolds and limits of Kleinian groups II
Troels Jørgensen conjectured that the algebraic and geometric limits of an algebraically convergent sequence of isomorphic Kleinian groups agree if there are no new parabolics in the algebraic limit. We prove that this conjecture holds in ‘most’ cases. In particular, we show that it holds when the domain of discontinuity of the algebraic limit of such a sequence is non-empty. We further show, with the same assumptions, that the limit sets of the groups in the sequence converge to the limit set of the algebraic limit. As a corollary, we verify the conjecture for finitely generated Kleinian groups which are not (non-trivial) free products of surface groups and infinite cyclic groups. These results are extensions of similar results for purely loxodromic groups. Thurston previously established these results in the case when the Kleinian groups are freely indecomposable. Using different techniques from ours, Ohshika has proven versions of these results for purely loxodromic function groups.
0024-6107
489-505
Anderson, James W.
739c0e33-ef61-4502-a675-575d08ee1a98
Canary, Richard D.
a26b9c05-b5d3-4837-a48e-0e7bb788c0b9
Anderson, James W.
739c0e33-ef61-4502-a675-575d08ee1a98
Canary, Richard D.
a26b9c05-b5d3-4837-a48e-0e7bb788c0b9

Anderson, James W. and Canary, Richard D. (2000) Cores of hyperbolic 3-manifolds and limits of Kleinian groups II. Journal of the London Mathematical Society, 61 (2), 489-505. (doi:10.1112/S0024610799008595).

Record type: Article

Abstract

Troels Jørgensen conjectured that the algebraic and geometric limits of an algebraically convergent sequence of isomorphic Kleinian groups agree if there are no new parabolics in the algebraic limit. We prove that this conjecture holds in ‘most’ cases. In particular, we show that it holds when the domain of discontinuity of the algebraic limit of such a sequence is non-empty. We further show, with the same assumptions, that the limit sets of the groups in the sequence converge to the limit set of the algebraic limit. As a corollary, we verify the conjecture for finitely generated Kleinian groups which are not (non-trivial) free products of surface groups and infinite cyclic groups. These results are extensions of similar results for purely loxodromic groups. Thurston previously established these results in the case when the Kleinian groups are freely indecomposable. Using different techniques from ours, Ohshika has proven versions of these results for purely loxodromic function groups.

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Published date: 2000

Identifiers

Local EPrints ID: 29871
URI: https://eprints.soton.ac.uk/id/eprint/29871
ISSN: 0024-6107
PURE UUID: 49327383-0f14-4486-823a-896bfa132650
ORCID for James W. Anderson: ORCID iD orcid.org/0000-0002-7849-144X

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Date deposited: 20 Jul 2006
Last modified: 06 Jun 2018 13:03

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