Gerasimov, Victor Expansive convergence groups are relatively hyperbolic. (English) Zbl 1226.20037 Geom. Funct. Anal. 19, No. 1, 137-169 (2009). 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. Cited in 1 ReviewCited in 21 Documents 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 Keywords:actions by homeomorphisms; properly discontinuous actions; geometrically finite actions; geometrically finite convergence groups; relatively hyperbolic groups; parabolic subgroups Citations:Zbl 0909.30034; Zbl 1043.20020 PDFBibTeX XMLCite \textit{V. Gerasimov}, Geom. Funct. Anal. 19, No. 1, 137--169 (2009; Zbl 1226.20037) Full Text: DOI