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Packing subgroups in relatively hyperbolic groups

G Christopher Hruska and Daniel T Wise

Geometry & Topology 13 (2009) 1945–1988

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.

relative hyperbolicity, quasiconvex subgroup, width, cube complex
Mathematical Subject Classification 2000
Primary: 20F65
Secondary: 20F67, 20F69
Received: 11 September 2006
Revised: 24 March 2009
Accepted: 7 February 2008
Published: 21 April 2009
Proposed: Martin Bridson
Seconded: Walter Neumann, Benson Farb
G Christopher Hruska
Department of Mathematical Sciences
University of Wisconsin–Milwaukee
PO Box 413
Milwaukee, WI 53201
Daniel T Wise
Department of Mathematics and Statistics
McGill University
Montreal, Quebec H3A 2K6