Packing subgroups in relatively hyperbolic groups

Abstract

We introduce the bounded packing property for a subgroup of a countable discrete group G . This property gives a finite upper bound on the number of left cosets of the subgroup that are pairwise close in G . We establish basic properties of bounded packing and give many examples; for instance, every subgroup of a countable, virtually nilpotent group has bounded packing. We explain several natural connections between bounded packing and group actions on CAT.0/ cube complexes.Our main result establishes the bounded packing of relatively quasiconvex subgroups of a relatively hyperbolic group, under mild hypotheses. As an application, we prove that relatively quasiconvex subgroups have finite height and width, properties that strongly restrict the way families of distinct conjugates of the subgroup can intersect. We prove that an infinite, nonparabolic relatively quasiconvex subgroup of a relatively hyperbolic group has finite index in its commensurator. We also prove a virtual malnormality theorem for separable, relatively quasiconvex subgroups, which is new even in the word hyperbolic case. 20F65; 20F67, 20F69

A combination theorem for strong relative hyperbolicity

“…All three theorems should have analogues in the (strongly) relatively hyperbolic world. Hruska and Wise [15] have already shown the finiteness of height and width of quasiconvex subgroups of relatively hyperbolic groups.…”

Quasi‐isometric groups with no common model geometry

“…The vertex and edge groups of the JSJ decomposition of are quasi‐convex, and hence satisfy the bounded packing property , introduced by Hruska–Wise . Definition Let be a finitely generated group and a word metric on defined with respect to a finite generating set.…”

“…The vertex and edge groups of the JSJ decomposition of G ∈ C k are quasi-convex, and hence satisfy the bounded packing property, introduced by Hruska-Wise [16].…”

Commensurated subgroups and ends of groups

“…In Section 6, we examine a connection between a left Schreier coset graph and the bounded packing ideas of C. Hruska and D. Wise [11].…”

“…As the cardinality of the set of ends of a graph is a quasi-isometry invariant, the graphs ƒ.¹x; tº; Q; G/ and Qn.S; G/ are not quasi-isometric. The main theorem of [11] is the following bounded packing result (which is more general and more sophisticated than Theorem 3.15):…”