an:05575932
Zbl 1226.20037
Gerasimov, Victor
Expansive convergence groups are relatively hyperbolic
EN
Geom. Funct. Anal. 19, No. 1, 137-169 (2009).
00250702
2009
j
20F67 20F65 30F40 57M07 57S05
actions by homeomorphisms; properly discontinuous actions; geometrically finite actions; geometrically finite convergence groups; relatively hyperbolic groups; parabolic subgroups
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 \textit{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 \textit{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.
Zbl 0909.30034; Zbl 1043.20020