an:01000456 Zbl 0873.20025 Kapovich, Ilya; Short, Hamish Greenberg's theorem for quasiconvex subgroups of word hyperbolic groups EN Can. J. Math. 48, No. 6, 1224-1244 (1996). 00037875 1996
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20F65 20E07 57M07 word hyperbolic groups; Gromov hyperbolic groups; boundaries of groups; quasi-convex subgroups; subgroups of finite index; virtual normalizers; geometrically finite groups of isometries; limit sets; commensurability The authors prove the following Theorem 1. Let $$G$$ be a word hyperbolic group, and let $$A$$ be an infinite quasi-convex subgroup of $$G$$. (i) If $$B$$ is an infinite quasi-convex subgroup of $$G$$ and $$A\cap B$$ has finite index in $$A$$ and in $$B$$, then $$A\cap B$$ has finite index in the subgroup generated by $$A$$ and $$B$$ in $$G$$. (ii) The subgroup $$A$$ has finite index in only finitely many subgroups of $$G$$. (iii) The subgroup $$A$$ has finite index in its virtual normalizer $$VN_G(A)$$, defined as $$VN_G(A)=\{g\in G\mid[A:A\cap gAg^{-1}]<\infty$$ and $$[gAg^{-1}:A\cap gAg^{-1}]<\infty\}$$. Part (ii) of this theorem is an analogue of a theorem of L. Greenberg on finitely generated subgroups of Fuchsian groups. As an application of the methods used, the authors give a proof of the following theorem due to G. Swarup: Theorem 2. Let $$G$$ be a torsion-free geometrically finite group of isometries of hyperbolic $$n$$-space with no parabolics. Then $$G$$ is word hyperbolic and a subgroup $$A$$ of $$G$$ is quasi-convex in $$G$$ if and only if it is geometrically finite. The authors obtain a criterion of commensurability for quasi-convex subgroups of word hyperbolic groups in terms of limit sets. They prove also the following theorem, which has also been obtained by M. Mihalik and W. Towle: Theorem 3. Let $$A$$ be an infinite quasi-convex subgroup of a word hyperbolic group $$G$$ and let $$B$$ be a subgroup of $$G$$ containing $$A$$. Then $$A$$ has finite index in $$B$$ if and only if $$A$$ contains an infinite subgroup $$C$$ which is normal in $$B$$. A.Papadopoulos (Strasbourg)