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Relatively hyperbolic groups. (English) Zbl 1259.20052
The author develops some of the foundations of the theory of relatively hyperbolic groups as originally formulated by M. Gromov in Chapter 8.6 of his article entitled Hyperbolic groups, [Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)]. Here is proved the equivalence of two definitions of this notion. One is essentially that of a group admitting a properly discontinuous geometrically finite action on a proper hyperbolic space. The other is that of a group which admits a cofinite action on a connected fine hyperbolic graph.
In the article is given a definition of a fine graph, i.e. it has only finitely many circuits of a given length containing any given edge. Moreover, there is shown how a relatively hyperbolic group can be assumed to act on a proper hyperbolic space of a particular geometric form.
Finally the author defines the boundary of a relatively hyperbolic group, and shows that the limit set of any geometrically finite action of the group is equivariantly homeomorphic to this boundary.
We should add that this very outstanding and valuable article was known since 1997 under the same title as a preprint of the author edited by the University of Southampton.

MSC:
20F67 Hyperbolic groups and nonpositively curved groups
20F65 Geometric group theory
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
57M07 Topological methods in group theory
53A35 Non-Euclidean differential geometry
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