On Quasiconvex Subsets of Hyperbolic Groups

Abstract

A geodesic metric space $X$ is called hyperbolic if there exists $delta ge 0$ such that every geodesic triangle $Delta$ in $X$ is $delta$-slim, i.e., each side of $Delta$ is contained in a closed $delta$-neighborhood of the two other sides. Let $G$ be a group generated by a finite set $A$ and let $Gamma$ be the corresponding Cayley graph. The group $G$ is said to be word hyperbolic if $Gamma$ is a hyperbolic metric space. A subset $Q$ of the group $G$ is called quasiconvex if for any geodesic $gamma$ connecting two elements from $Q$ in $Gamma$, $gamma$ is contained in a closed $varepsilon$-neighborhood of $Q$ (for some fixed $varepsilon ge 0$). Quasiconvex subgroups play an important role in the theory of hyperbolic groups and have been studied quite thoroughly. We investigate properties of quasiconvex subsets in word hyperbolic groups and generalize a number of results previously known about quasiconvex subgroups. On the other hand, we establish and study a notion of quasiconvex subsets that are small relatively to subgroups. This allows to prove a theorem concerning residualizing homomorphisms preserving such subsets. As corollaries, we obtain several new embedding theorems for word hyperbolic groups.