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Convergence groups and configuration spaces. (English) Zbl 0952.20032
Cossey, John (ed.) et al., Geometric group theory down under. Proceedings of a special year in geometric group theory, Canberra, Australia, July 14-19, 1996. Berlin: de Gruyter. 23-54 (1999).
The paper develops some of the basic properties of convergence groups (initially introduced by F. W. Gehring and G. J. Martin [Lect. Notes Math. 1275, 158-167 (1987; Zbl 0623.30030)]) in the fairly general context of an arbitrary compact Hausdorff space, from the point of view of the induced action on the space of distinct triples. This view is equivalent to the original Gehring-Martin definition in the case of topological spheres – it axiomatises the essential dynamical properties of a discrete conformal action on the ideal sphere of real hyperbolic space, see the reviewer’s book [Conformal geometry of discrete groups and manifolds, Walter de Gruyter (2000)]. The motivation of this generalization stems from the fact that a word hyperbolic group (in the sense of Gromov) acting on its boundary satisfies the convergence axioms.
For the entire collection see [Zbl 0910.00040].

##### MSC:
 20F65 Geometric group theory 57M60 Group actions on manifolds and cell complexes in low dimensions 57M07 Topological methods in group theory 20F67 Hyperbolic groups and nonpositively curved groups 54F50 Topological spaces of dimension $$\leq 1$$; curves, dendrites