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Boundaries of geometrically finite groups

Boundaries of geometrically finite groups
Boundaries of geometrically finite groups
We show that the limit set of a relatively hyperbolic group with no separating horoball is locally connected if it is connected. On the other hand, if there is a separating horoball centred on a parabolic point, one obtains a non-trivial splitting of the group over a parabolic subgroup relative to the maximal parabolic subgroups. Together with results from elsewhere, one deduces that if $ \Gamma $ is a relatively hyperbolic group such that each maximal parabolic subgroup is one-or-two ended, finitely presented, and contains no infinite torsion subgroup, then the boundary of $ \Gamma $ is locally connected if it is connected. As a corollary, we see that the limit set of a geometrically finite group acting on a complete simply connected manifold of pinched negative curvature must be locally connected if it is connected.
0025-5874
509-527
Bowditch, B.H.
8f3cf0c9-0a10-4b70-8648-33fb2ade7ac9
Bowditch, B.H.
8f3cf0c9-0a10-4b70-8648-33fb2ade7ac9

Bowditch, B.H. (1999) Boundaries of geometrically finite groups. Mathematische Zeitschrift, 230 (3), 509-527.

Record type: Article

Abstract

We show that the limit set of a relatively hyperbolic group with no separating horoball is locally connected if it is connected. On the other hand, if there is a separating horoball centred on a parabolic point, one obtains a non-trivial splitting of the group over a parabolic subgroup relative to the maximal parabolic subgroups. Together with results from elsewhere, one deduces that if $ \Gamma $ is a relatively hyperbolic group such that each maximal parabolic subgroup is one-or-two ended, finitely presented, and contains no infinite torsion subgroup, then the boundary of $ \Gamma $ is locally connected if it is connected. As a corollary, we see that the limit set of a geometrically finite group acting on a complete simply connected manifold of pinched negative curvature must be locally connected if it is connected.

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Published date: 1999

Identifiers

Local EPrints ID: 29772
URI: http://eprints.soton.ac.uk/id/eprint/29772
ISSN: 0025-5874
PURE UUID: f752ea60-2a98-48e1-952e-40c165c59b46

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Date deposited: 27 Jul 2006
Last modified: 15 Jul 2019 19:08

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