On quasiconvex subsets of hyperbolic groups

A geodesic metric space X is called hyperbolic if there exists delta ≥ 0 such that every geodesic triangle ▵ in X is delta-slim, i.e., each side of ▵ 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(G, A ) be the corresponding Cayley graph. The group G is said to be word hyperbolic if Gamma(G, A ) 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(G, A ), gamma is contained in a closed epsilon-neighborhood of Q (for some fixed epsilon ≥ 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.