Volume 13, issue 4 (2009)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 20
Issue 4, 1807–2438
Issue 3, 1257–1806
Issue 2, 629–1255
Issue 1, 1–627

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Subscriptions
Author Index
To Appear
Contacts
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Packing subgroups in relatively hyperbolic groups

G Christopher Hruska and Daniel T Wise

Geometry & Topology 13 (2009) 1945–1988
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.

Keywords
relative hyperbolicity, quasiconvex subgroup, width, cube complex
Mathematical Subject Classification 2000
Primary: 20F65
Secondary: 20F67, 20F69
References
Publication
Received: 11 September 2006
Revised: 24 March 2009
Accepted: 7 February 2008
Published: 21 April 2009
Proposed: Martin Bridson
Seconded: Walter Neumann, Benson Farb
Authors
G Christopher Hruska
Department of Mathematical Sciences
University of Wisconsin–Milwaukee
PO Box 413
Milwaukee, WI 53201
USA
http://www.uwm.edu/~chruska
Daniel T Wise
Department of Mathematics and Statistics
McGill University
Montreal, Quebec H3A 2K6
Canada
http://www.math.mcgill.ca/~wise