Expansive convergence groups are relatively hyperbolic.

*(English)*Zbl 1226.20037Summary: 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 |