We lay the foundations for the study of relatively quasiconvex subgroups of relatively hyperbolic groups. These foundations require that we first work out a coherent theory of countable relatively hyperbolic groups (not necessarily finitely generated). We prove the equivalence of Gromov, Osin and Bowditch's definitions of relative hyperbolicity for countable groups.We then give several equivalent definitions of relatively quasiconvex subgroups in terms of various natural geometries on a relatively hyperbolic group. We show that each relatively quasiconvex subgroup is itself relatively hyperbolic, and that the intersection of two relatively quasiconvex subgroups is again relatively quasiconvex. In the finitely generated case, we prove that every undistorted subgroup is relatively quasiconvex, and we compute the distortion of a finitely generated relatively quasiconvex subgroup. 20F65, 20F67

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

Abstract. We give a generalized and self-contained account of Haglund-Paulin's wallspaces and Sageev's construction of the CAT(0) cube complex dual to a wallspace. We examine criteria on a wallspace leading to finiteness properties of its dual cube complex. Our discussion is aimed at readers wishing to apply these methods to produce actions of groups on cube complexes and understand their nature. We develop the wallspace ideas in a level of generality that facilitates their application.Our main result describes the structure of dual cube complexes arising from relatively hyperbolic groups. Let H1, . . . , Hs be relatively quasiconvex codimension-1 subgroups of a group G that is hyperbolic relative to P1, . . . , Pr. We prove that G acts relatively cocompactly on the associated dual CAT(0) cube complex C. This generalizes Sageev's result that C is cocompact when G is hyperbolic. When P1, . . . , Pr are abelian, we show that the dual CAT(0) cube complex C has a G-cocompact CAT(0) truncation.

The purpose of this erratum is to correct the proof of Theorem A.0.1 in the appendix to [4], which was jointly authored by Mohamad Hindawi, Hruska and Kleiner. In that appendix, many of the results of [4] about CAT(0) spaces with isolated flats are extended to a more general setting in which the isolated subspaces are not necessarily flats. However, one step of that extension does not follow from the argument we used the isolated flats setting. We provide a new proof that fills this gap.In addition, we give a more detailed account of several other parts of Theorem A.0.1, which were sketched in [4].

We study those groups that act properly discontinuously, cocompactly, and isometrically on CAT(0) spaces with isolated flats and the Relative Fellow Traveller Property. The groups in question include word hyperbolic CAT(0) groups as well as geometrically finite Kleinian groups and numerous 2-dimensional CAT(0) groups. For such a group we show that there is an intrinsic notion of a quasiconvex subgroup which is equivalent to the inclusion being a quasi-isometric embedding. We also show that the visual boundary of the CAT(0) space is actually an invariant of the group. More generally, we show that each quasiconvex subgroup of such a group has a canonical limit set which is independent of the choice of overgroup.The main results in this article were established by Gromov and Short in the word hyperbolic setting and do not extend to arbitrary CAT(0) groups.