Skip to main content Accessibility help


  • Eduardo Martínez-Pedroza (a1) and Piotr Przytycki (a2)


Let $G$ be a group hyperbolic relative to a finite collection of subgroups ${\mathcal{P}}$ . Let ${\mathcal{F}}$ be the family of subgroups consisting of all the conjugates of subgroups in ${\mathcal{P}}$ , all their subgroups, and all finite subgroups. Then there is a cocompact model for $E_{{\mathcal{F}}}G$ . This result was known in the torsion-free case. In the presence of torsion, a new approach was necessary. Our method is to exploit the notion of dismantlability. A number of sample applications are discussed.



Hide All
1. Alonso, J. M., Brady, T., Cooper, D., Ferlini, V., Lustig, M., Mihalik, M., Shapiro, M. and Short, H., Notes on word hyperbolic groups, in Group Theory from a Geometrical Viewpoint (Trieste, 1990), pp. 363 (World Scientific Publishing, River Edge, NJ, 1991). Edited by Short.
2. Barmak, J. A. and Minian, E. G., Strong homotopy types, nerves and collapses, Discrete Comput. Geom. 47(2) (2012), 301328.
3. Bowditch, B. H., Relatively hyperbolic groups, Int. J. Algebra Comput. 22(3) (2012), 12500161250066.
4. Bredon, G. E., Equivariant Cohomology Theories, Lecture Notes in Mathematics, volume 34 (Springer, Berlin–New York, 1967).
5. Bridson, M. R. and Haefliger, A., Metric Spaces of Non-positive Curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], volume 319, (Springer, Berlin, 1999).
6. Chepoi, V. and Osajda, D., Dismantlability of weakly systolic complexes and applications, Trans. Amer. Math. Soc. 367(2) (2015), 12471272.
7. Dahmani, F., Les groupes relativement hyperboliques et leurs bords, PhD thesis (2003).
8. Dahmani, F., Classifying spaces and boundaries for relatively hyperbolic groups, Proc. Lond. Math. Soc. (3) 86(3) (2003), 666684.
9. Druţu, C. and Sapir, M. V., Groups acting on tree-graded spaces and splittings of relatively hyperbolic groups, Adv. Math. 217(3) (2008), 13131367.
10. Fletcher, J. L., Homological Group Invariants, PhD thesis (1998).
11. Gersten, S. M., Subgroups of word hyperbolic groups in dimension 2, J. Lond. Math. Soc. (2) 54(2) (1996), 261283.
12. Hanlon, R. G. and Martínez-Pedroza, E., A subgroup theorem for homological filling functions, Groups Geom. Dyn. 10(3) (2016), 867883.
13. Hensel, S., Osajda, D. and Przytycki, P., Realisation and dismantlability, Geom. Topol. 18(4) (2014), 20792126.
14. Hruska, G. C., Relative hyperbolicity and relative quasiconvexity for countable groups, Algebr. Geom. Topol. 10(3) (2010), 18071856.
15. Lang, U., Injective hulls of certain discrete metric spaces and groups, J. Topol. Anal. 5(3) (2013), 297331.
16. Lück, W., Transformation Groups and Algebraic K-theory, Lecture Notes in Mathematics, volume 1408 (Springer, Berlin, 1989). Mathematica Gottingensis.
17. Lück, W. and Meintrup, D., On the universal space for group actions with compact isotropy, in Geometry and Topology: Aarhus (1998), Contemporary Mathematics, volume 258, pp. 293305 (American Mathematical Society, Providence, RI, 2000).
18. Martínez-Pedroza, E., Subgroups of relatively hyperbolic groups of Bredon cohomological dimension 2, preprint, 2015, arXiv:1508.04865.
19. Martínez-Pedroza, E., A note on fine graphs and homological isoperimetric inequalities, Canad. Math. Bull. 59(1) (2016), 170181.
20. Martínez-Pedroza, E. and Wise, D. T., Relative quasiconvexity using fine hyperbolic graphs, Algebr. Geom. Topol. 11(1) (2011), 477501.
21. Meintrup, D. and Schick, T., A model for the universal space for proper actions of a hyperbolic group, New York J. Math. 8(1–7) (2002), (electronic).
22. Mineyev, I. and Yaman, A., Relative hyperbolicity and bounded cohomology, Available at mineyev/math/art/rel-hyp.pdf, 2007.
23. Osin, D. V., Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems, Mem. Amer. Math. Soc. 179(843) (2006), vi+100.
24. Osin, D. V., Peripheral fillings of relatively hyperbolic groups, Invent. Math. 167(2) (2007), 295326.
25. Polat, N., Finite invariant simplices in infinite graphs, Period. Math. Hungar. 27(2) (1993), 125136.
26. Segev, Y., Some remarks on finite 1-acyclic and collapsible complexes, J. Combin. Theory Ser. A 65(1) (1994), 137150.
27. tom Dieck, T., Transformation Groups, de Gruyter Studies in Mathematics, volume 8 (Walter de Gruyter & Co., Berlin, 1987).
MathJax is a JavaScript display engine for mathematics. For more information see


MSC classification


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed