Quasiconvex subgroups of negatively curved groups (Q1336794)

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.