A non-quasiconvexity embedding theorem for hyperbolic groups. (English) Zbl 0942.20026

The author considers quasiconvex subgroups, and we recall the definition. A subset \(Y\) of a metric space \(X\) is said to be (\(\epsilon\))-quasiconvex if for every pair of points in \(Y\), any geodesic segment joining them is contained in the \(\epsilon\)-neighborhood of \(Y\). Let \(G\) be now a hyperbolic group, equipped with a finite generating set, let \(K\) be the corresponding Cayley graph equipped with its word metric and let \(A\) be a subgroup of \(G\). Then \(A\) is said to be a quasiconvex subgroup of \(G\) if it is quasiconvex as a subset of the metric space \(K\).
The main result of this paper is the following Theorem A. If \(G\) is a not virtually cyclic torsion free hyperbolic group then there exists another word hyperbolic group \(G^*\) such that \(G\) is a subgroup of \(G^*\) but not quasiconvex in \(G^*\). – As the author points out, examples of finitely generated subgroups of hyperbolic groups that are not quasiconvex were rare.
Theorem B. Let \(G\) be a torsion-free hyperbolic group and let \(\Gamma\) be a non-cyclic subgroup of \(G\). Then, there exists a subgroup \(H\) of \(\Gamma\) such that \(H\) is a free group of rank two which is quasiconvex and malnormal in \(G\) (meaning that for any \(g\in G-H\) we have \(H\cap g^{-1}Hg=1\)). – The author discusses also a parallel between quasiconvexity and geometric finiteness for a group.


20F67 Hyperbolic groups and nonpositively curved groups
20F65 Geometric group theory
57M07 Topological methods in group theory
20E07 Subgroup theorems; subgroup growth
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