## Greenberg’s theorem for quasiconvex subgroups of word hyperbolic groups.(English)Zbl 0873.20025

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 $$B$$.

### MSC:

 20F65 Geometric group theory 20E07 Subgroup theorems; subgroup growth 57M07 Topological methods in group theory
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