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Limit sets of geometrically finite hyperbolic groups. (English) Zbl 0791.30039
In this paper the authors study the existing relationship between geometrically finite hyperbolic groups in dimension \(n\) and their limit sets. Also they extend to dimension \(n\) several results which had already been established for dimension 2 and 3. Thus they prove: Let \(H \subseteq G\) be hyperbolic groups with \(H\) nonelementary and geometrically finite. If the limit sets \(\Delta (H)\) and \(\Delta (G)\) are equal, then the index of \(H\) in \(G\) is finite. Also they know that the intersection of two geometrically finite subgroups of a hyperbolic group is geometrically finite. Some properties of limit sets of intersection of two subgroups of a hyperbolic group are also studied.

MSC:
30F99 Riemann surfaces
57N10 Topology of general \(3\)-manifolds (MSC2010)
51M10 Hyperbolic and elliptic geometries (general) and generalizations
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