Mathematics on Quasiconvex Subsets of Hyperbolic Groups

A geodesic metric space X is called hyperbolic if there exists δ ≥ 0 such that every geodesic triangle ∆ in X is δ-slim, i.e., each side of ∆ is contained in a closed δ-neighborhood of the two other sides. Let G be a group generated by a finite set A and let Γ(G,A) be the corresponding Cayley graph. The group G is said to be word hyperbolic if Γ(G,A) is a hyperbolic metric space. A subset Q of the group G is called quasiconvex if for any geodesic γ connecting two elements from Q in Γ(G,A), γ is contained in a closed ε-neighborhood of Q (for some fixed ε ≥ 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.