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Expansive convergence groups are relatively hyperbolic. (English) Zbl 1226.20037
Summary: 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 [J. Reine Angew. Math. 501, 71-98 (1998; Zbl 0909.30034)]. So, we have another topological characterisation of geometrically finite convergence groups and, by the result of A. Yaman [J. Reine Angew. Math. 566, 41-89 (2004; Zbl 1043.20020)], of relatively hyperbolic groups. Further, if \(G\) is finitely generated then the parabolic subgroups are finitely generated and undistorted. This answers a question of B. Bowditch and eliminates restrictions in some known theorems about relatively hyperbolic groups.

MSC:
20F67 Hyperbolic groups and nonpositively curved groups
20F65 Geometric group theory
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
57M07 Topological methods in group theory
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
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