Greenberg’s theorem for quasiconvex subgroups of word hyperbolic groups.

*(English)*Zbl 0873.20025The 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 \(B\).

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 \(B\).

Reviewer: A.Papadopoulos (Strasbourg)

##### MSC:

20F65 | Geometric group theory |

20E07 | Subgroup theorems; subgroup growth |

57M07 | Topological methods in group theory |