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Quasiconvex subgroups of negatively curved groups. (English) Zbl 0822.20038
The authors prove the following theorem: Let $$H$$ be a negatively curved group (that is, a hyperbolic group in the sense of Gromov) and let $$A$$ be an infinite quasi-convex subgroup of $$H$$. Then: (1) $$A$$ has finite index in the normalizer of $$A$$ in $$H$$. (2) If $$h\in H$$ and $$hAh^{-1}$$ is a subset of $$A$$, then $$hAh^{-1} = A$$. (3) If $$N$$ is an infinite normal subgroup of $$H$$ and $$N\subset A$$, then $$A$$ has finite index in $$H$$.
Special cases of this theorem were already known, and the proof uses the basic hyperbolic group techniques.

##### MSC:
 20F65 Geometric group theory 20F05 Generators, relations, and presentations of groups 20E07 Subgroup theorems; subgroup growth
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##### References:
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