  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.
