Relatively hyperbolic groups
Relatively hyperbolic groups
In this paper, we develop some of the foundations of the theory of relatively hyperbolic groups, as originally formulated by Gromov. We prove the equivalence of two definitions of this notion. One is essentially that of a group admitting a properly discontinuous geometrically finite action on a proper hyperbolic space, that is, such that every limit point is either a conical limit point or a bounded parabolic point. The other is that of a group which admits a cofinite action on a connected fine hyperbolic graph. We define a graph to be "fine" if there are only finitely many circuits a given length containing any given edge, and we develop some of the properties of this notion. We show how a relatively hyperbolic group can be assumed to act on a proper hyperbolic space of a particular geometric form. We define the boundary of a relatively hyperbolic group, and show that the limit set of any geometrically finite action of the group is equivariantly homeomorphic to this boundary. This generalises a result of Tukia for geometrically finite kleinian groups.
63pp
Bowditch, B.H.
8f3cf0c9-0a10-4b70-8648-33fb2ade7ac9
1997
Bowditch, B.H.
8f3cf0c9-0a10-4b70-8648-33fb2ade7ac9
Bowditch, B.H.
(1997)
Relatively hyperbolic groups.
Preprint, .
Abstract
In this paper, we develop some of the foundations of the theory of relatively hyperbolic groups, as originally formulated by Gromov. We prove the equivalence of two definitions of this notion. One is essentially that of a group admitting a properly discontinuous geometrically finite action on a proper hyperbolic space, that is, such that every limit point is either a conical limit point or a bounded parabolic point. The other is that of a group which admits a cofinite action on a connected fine hyperbolic graph. We define a graph to be "fine" if there are only finitely many circuits a given length containing any given edge, and we develop some of the properties of this notion. We show how a relatively hyperbolic group can be assumed to act on a proper hyperbolic space of a particular geometric form. We define the boundary of a relatively hyperbolic group, and show that the limit set of any geometrically finite action of the group is equivariantly homeomorphic to this boundary. This generalises a result of Tukia for geometrically finite kleinian groups.
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Published date: 1997
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Paper revised, March 1999.
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Local EPrints ID: 29769
URI: http://eprints.soton.ac.uk/id/eprint/29769
PURE UUID: f9fa8073-926c-4459-8f7b-212931a402e2
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Date deposited: 02 May 2007
Last modified: 15 Mar 2024 07:34
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Author:
B.H. Bowditch
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