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On quasiconvex subgroups of word hyperbolic groups. (English) Zbl 0994.20036
The author studies properties of quasi-convex subgroups of word hyperbolic groups. The main result is the following Theorem: Let \(G\) be a non-elementary torsion-free word hyperbolic group and let \(H\) be a quasiconvex subgroup of \(G\) of finite index. Then there exists a non-trivial element \(g\in G\) such that the subgroup generated by \(H\) and \(g\) is the free product \(H*\langle g\rangle\) and is quasiconvex in \(G\).
The theorem was already stated by M. Gromov in his paper [Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)], Section 5.3.C, with a sketch of a proof.

20F67 Hyperbolic groups and nonpositively curved groups
20E07 Subgroup theorems; subgroup growth
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F65 Geometric group theory
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