zbMATH — the first resource for mathematics

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.

20F65 Geometric group theory
20F05 Generators, relations, and presentations of groups
20E07 Subgroup theorems; subgroup growth
Full Text: DOI
[1] J.M. Alonso and M.R. Bridson, Semihyperbolic groups, J. London Math. Soc., to appear. · Zbl 0823.20035
[2] Coornaert, M.; Delzant, T.; Papadopoulous, A., Notes sur LES groupes hyperboliques de Gromov, ()
[3] Gersten, S.M.; Short, H., Rational subgroups of biautomatic groups, Ann. of math., 134, 125-158, (1991) · Zbl 0744.20035
[4] Hyperbolic groups, (), 3-63 · Zbl 0849.20023
[5] Sur LES groupes hyperboliques d’après mikhael Gromov, () · Zbl 0731.20025
[6] Gromov, M., Hyperbolic groups, () · Zbl 0634.20015
[7] Magnus, W.; Karrass, A.; Solitar, D., Combinatorial group theory, (1976), Dover New York
[8] Neumann, W.D., Asynchronous combings of groups, Internat J. algebra and comput., 2, 2, 179-185, (1992) · Zbl 0777.20013
[9] Short, H., Groups and combings, (1990), ENS de Lyon, Preprint
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.