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Quasiconvexity in relatively hyperbolic groups. (English) Zbl 1361.20031
The authors study different notions of quasiconvexity for a subgroup \(H\) of a relatively hyperbolic group \(G\). It follows from the first main result of the paper under review that relative geometric quasiconvexity is equivalent to dynamical quasiconvexity as was conjectured in [D. V. Osin, Mem. Am. Math. Soc. 843, 100 p. (2006; Zbl 1093.20025)]. The second main result claims that, for a finitely generated \(G\), the action of \(H\) outside its limit set is cocompact if and only if it is absolutely quasiconvex and every infinite intersection of \(H\) with a parabolic subgroup of \(G\) has finite index in the parabolic subgroup.

MSC:
20F67 Hyperbolic groups and nonpositively curved groups
20F65 Geometric group theory
20E07 Subgroup theorems; subgroup growth
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