On Products of Quasiconvex Subgroups in Hyperbolic Groups

An interesting question about quasiconvexity in a hyperbolic group concerns finding classes of quasiconvex subsets that are closed under finite intersections. A known example is the class of all quasiconvex subgroups [1]. However, not much is yet learned about the structure of arbitrary quasiconvex subsets. In this work we study the properties of products of quasiconvex subgroups; we show that such sets are quasiconvex, their finite intersections have a similar algebraic representation and, thus, are quasiconvex too. 0. Introduction Let G be a hyperbolic group, Γ(G,A) – its Cayley graph corresponding to a finite symmetrized generating set A (i.e. for each element a ∈ A, a also belongs to this set). A subset Q ⊆ G is said to be ε-quasiconvex, if any geodesic connecting two elements from Q belongs to a closed ε-neighborhood Oε(Q) of Q in Γ(G,A) for some ε ≥ 0. Q will be called quasiconvex if there exists ε > 0 for which it is ε-quasiconvex. In [4] Gromov proves that the notion of quasiconvexity in a hyperbolic group does not depend on the choice of a finite generating set (it is easy to show that this is not true in an arbitrary group). If A,B ⊆ G then their product is a subset of G defined by A · B = {ab | a ∈ A, b ∈ B}. Proposition 1. If the sets A1, . . . , An ⊂ G are quasiconvex then their product set A1A2 · . . . ·An def = {a1a2 · . . . · an | ai ∈ Gi} ⊂ G is also quasiconvex. Proposition 1 was proved by Zeph Grunschlag in 1999 in [11; Prop. 3.14] and, independently, by the author in his diploma paper in 2000. If H is a subgroup of G and x ∈ G then the subgroup conjugated to H by x will be denoted H = xHx. The main result of the paper is Theorem 1. Suppose G1, . . . , Gn, H1, . . . , Hm are quasiconvex subgroups of the group G, n,m ∈ N; f, e ∈ G. Then there exist numbers r, tl ∈ N ∪ {0} and fl, αlk, βlk ∈ G, k = 1, 2, . . . , tl (for every fixed l), l = 1, 2, . . . , r, such that fG1G2 · . . . ·Gn ∩ eH1H2 · . . . ·Hm = r