Relative quasiconvexity using fine hyperbolic graphs

Abstract

We provide a new and elegant approach to relative quasiconvexity for relatively hyperbolic groups in the context of Bowditch's approach to relative hyperbolicity using cocompact actions on fine hyperbolic graphs. Our approach to quasiconvexity generalizes the other definitions in the literature that apply only for countable relatively hyperbolic groups. We also provide an elementary and self-contained proof that relatively quasiconvex subgroups are relatively hyperbolic. 20F06, 20F65, 20F67

On quasiconvexity and relatively hyperbolic structures on groups

“…[14] assuming that G is finitely generated, and the version for countable groups appeared in [8]. Definition 3.2 is equivalent to other approaches to relative quasiconvexity and we refer the interested reader to [8,12] and the references there in.…”

“…Theorem 3.7 [8,11] Let Q and R be quasiconvex subgroups of G relative to H. Then Q ∩ R is quasiconvex in G relative to H. Theorem 3.8 [8,12] Let Q be a σ -quasiconvex subgroup of G relative to H, and let L = {Q ∩ g Hg −1 : g ∈ G, H ∈ H, dist(1, g) ≤ σ }. Then Q is hyperbolic relative to L. In particular, Q is finitely generated relative to L, and each infinite maximal parabolic subgroup of Q with respect to H is conjugate in Q to an element of L…”

“…Equivalent notions of relatively quasiconvex subgroups for relatively hyperbolic groups have also been introduced by different authors and they have been proved to be equivalent, we refer the reader to [8,12].…”

“…The collection of quasiconvex subgroups of a (strong) relatively hyperbolic pair (G, H) is an interesting class of subgroups of G. For example, this class is closed under finite intersections [8,11], contains all virtually cyclic subgroups [14], and every element naturally admits a relatively hyperbolic group structure [8,12]. A countable group G can have different relatively hyperbolic structures giving rise to different collections of subgroups of G with these properties.…”

Notes on Relatively Hyperbolic Groups and Relatively Quasiconvex Subgroups

“…Suppose that L is strongly undistorted relative to H in G. We take a finite generating system S of L. By taking Y = ∅ ⊂ G, we recognize that L is undistorted relative to H in G. By the assertion (i), L is quasiconvex relative to H in G. Assume that the subgroup L ∩ gHg −1 is infinite for some g ∈ G and H ∈ H. The case where G is countable and H is finite was known (see [11,Corollary 3.5]). Theorem 4.25.…”