# zbMATH — the first resource for mathematics

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.

##### MSC:
 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
Full Text: