## Geometry & Topology

### 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.

#### Article information

Source
Geom. Topol., Volume 13, Number 4 (2009), 1945-1988.

Dates
Received: 11 September 2006
Revised: 24 March 2009
Accepted: 7 February 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513800283

Digital Object Identifier
doi:10.2140/gt.2009.13.1945

Mathematical Reviews number (MathSciNet)
MR2497315

Zentralblatt MATH identifier
1188.20042

#### Citation

Hruska, G Christopher; Wise, Daniel T. Packing subgroups in relatively hyperbolic groups. Geom. Topol. 13 (2009), no. 4, 1945--1988. doi:10.2140/gt.2009.13.1945. https://projecteuclid.org/euclid.gt/1513800283

#### References

• J M Alonso, T Brady, D Cooper, V Ferlini, M Lustig, M Mihalik, H Short (editor), Notes on word hyperbolic groups, from: “Group theory from a geometrical viewpoint (Trieste, 1990)”, (E Ghys, A Haefliger, A Verjovsky, editors), World Sci. Publ., River Edge, NJ (1991) 3–63
• H-J Bandelt, M van de Vel, Superextensions and the depth of median graphs, J. Combin. Theory Ser. A 57 (1991) 187–202
• B Bowditch, Relatively hyperbolic groups, Univ. of Southampton Preprint Ser. (1999)Http://www.warwick.ac.uk/~masgak/papers/bhb-relhyp.pdf
• V Chepoi, Graphs of some ${\rm CAT}(0)$ complexes, Adv. in Appl. Math. 24 (2000) 125–179
• C Dru\RomanianCommatu, M Sapir, Tree-graded spaces and asymptotic cones of groups, Topology 44 (2005) 959–1058 With an appendix by D Osin and Sapir
• B Farb, Relatively hyperbolic groups, Geom. Funct. Anal. 8 (1998) 810–840
• V N Gerasimov, Semi-splittings of groups and actions on cubings, from: “Algebra, geometry, analysis and mathematical physics (Novosibirsk, 1996)”, (Y G Reshetnyak, L A Bokut$'$, S K Vodop$'$yanov, I A Taĭmanov, editors), Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk (1997) 91–109, 190 In Russian; translated in Siberian Adv. Math. 8 (1998) 36–58
• R Gitik, M Mitra, E Rips, M Sageev, Widths of subgroups, Trans. Amer. Math. Soc. 350 (1998) 321–329
• M Gromov, Hyperbolic groups, from: “Essays in group theory”, (S M Gersten, editor), Math. Sci. Res. Inst. Publ. 8, Springer, New York (1987) 75–263
• G C Hruska, Relative hyperbolicity and relative quasiconvexity for countable groups
• G C Hruska, D Wise, Finiteness properties of cubulated groups, in preparation
• E Martínez-Pedroza, Combination of quasiconvex subgroups of relatively hyperbolic groups, Groups Geom. Dyn. 3 (2009) 317–342
• G A Niblo, L D Reeves, Coxeter groups act on ${\rm CAT}(0)$ cube complexes, J. Group Theory 6 (2003) 399–413
• G A Niblo, M A Roller, Groups acting on cubes and Kazhdan's property (T), Proc. Amer. Math. Soc. 126 (1998) 693–699
• D V Osin, Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems, Mem. Amer. Math. Soc. 179 (2006) vi+100
• E Rips, Subgroups of small cancellation groups, Bull. London Math. Soc. 14 (1982) 45–47
• M Roller, Poc sets, median algebras and group actions. An extended study of Dunwoody's construction and Sageev's theorem, Univ. of Southampton Preprint Ser. (1998)
• J H Rubinstein, M Sageev, Intersection patterns of essential surfaces in $3$–manifolds, Topology 38 (1999) 1281–1291
• J H Rubinstein, S Wang, $\pi\sb 1$–injective surfaces in graph manifolds, Comment. Math. Helv. 73 (1998) 499–515
• M Sageev, Ends of group pairs and non-positively curved cube complexes, Proc. London Math. Soc. $(3)$ 71 (1995) 585–617
• M Sageev, Codimension–$1$ subgroups and splittings of groups, J. Algebra 189 (1997) 377–389