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# Relatively hyperbolic groups

Record type: Article

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. 2000 Subject Classification : 20F57.

PDF bhb-relhyp.pdf - Author's Original

## Citation

Bowditch, B.H. (1997) Relatively hyperbolic groups Preprint, 63pp.

Published date: 1997
Additional Information: Paper revised, March 1999.

## Identifiers

Local EPrints ID: 29769
URI: http://eprints.soton.ac.uk/id/eprint/29769
PURE UUID: f9fa8073-926c-4459-8f7b-212931a402e2

## Catalogue record

Date deposited: 02 May 2007

## Contributors

Author: B.H. Bowditch