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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
##### Keywords:
hyperbolic group; discrete group; compactum
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