Abstract.
Let a discrete group G act by homeomorphisms of a compactum in a way that the action is properly discontinuous on triples and cocompact on pairs. We prove that such an action is geometrically finite. The converse statement was proved by P. Tukia [T3]. So, we have another topological characterisation of geometrically finite convergence groups and, by the result of A. Yaman [Y2], of relatively hyperbolic groups. Further, if G is finitely generated then the parabolic subgroups are finitely generated and undistorted. This answer to a question of B. Bowditch and eliminates restrictions in some known theorems about relatively hyperbolic groups.
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This work was completed during the 3-month visit to Laboratoire Paule Painlevé, Lille, France, supported by CNRS.
Received: April 2007, Revision: May 2008, Accepted: August 2008
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Gerasimov, V. Expansive Convergence Groups are Relatively Hyperbolic. Geom. Funct. Anal. 19, 137–169 (2009). https://doi.org/10.1007/s00039-009-0718-7
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DOI: https://doi.org/10.1007/s00039-009-0718-7