zbMATH — the first resource for mathematics

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.

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