Kyoto Journal of Mathematics

Extending properties to relatively hyperbolic groups

Daniel A. Ramras and Bobby W. Ramsey

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Consider a finitely generated group G that is relatively hyperbolic with respect to a family of subgroups H1,,Hn. We present an axiomatic approach to the problem of extending metric properties from the subgroups Hi to the full group G. We use this to show that both (weak) finite decomposition complexity and straight finite decomposition complexity are extendable properties. We also discuss the equivalence of two notions of straight finite decomposition complexity.

Article information

Kyoto J. Math., Volume 59, Number 2 (2019), 343-356.

Received: 12 August 2016
Revised: 17 February 2017
Accepted: 22 February 2017
First available in Project Euclid: 18 January 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F67: Hyperbolic groups and nonpositively curved groups
Secondary: 19D50: Computations of higher $K$-theory of rings [See also 13D15, 16E20] 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 20F69: Asymptotic properties of groups

relative hyperbolicity finite decomposition complexity asymptotic dimension Cayley graph


Ramras, Daniel A.; Ramsey, Bobby W. Extending properties to relatively hyperbolic groups. Kyoto J. Math. 59 (2019), no. 2, 343--356. doi:10.1215/21562261-2018-0017.

Export citation


  • [1] B. H. Bowditch, Relatively hyperbolic groups, Internat. J. Algebra Comput. 22 (2012), no. 3, art. ID 1250016.
  • [2] G. Carlsson and B. Goldfarb, Algebraic $K$-theory of geometric groups, preprint, arXiv:1305.3349v3 [math.AT].
  • [3] M. Dadarlat and E. Guentner, Uniform embeddability of relatively hyperbolic groups, J. Reine Angew. Math. 612 (2007), 1–15.
  • [4] A. Dranishnikov and M. Zarichnyi, Universal spaces for asymptotic dimension, Topology Appl. 140 (2004), nos. 2–3, 203–225.
  • [5] A. Dranishnikov and M. Zarichnyi, Asymptotic dimension, decomposition complexity, and Haver’s property C, Topology Appl. 169 (2014), 99–107.
  • [6] J. Dydak and Z. Virk, Preserving coarse properties, Rev. Mat. Complut. 29 (2016), no. 1, 191–206.
  • [7] B. Farb, Relatively hyperbolic groups, Geom. Funct. Anal. 8 (1998), no. 5, 810–840.
  • [8] T. Fukaya and S. Oguni, The coarse Baum-Connes conjecture for relatively hyperbolic groups, J. Topol. Anal. 4 (2012), no. 1, 99–113.
  • [9] B. Goldfarb, Weak coherence of groups and finite decomposition complexity, to appear in Int. Math. Res. Not. IMRN, preprint, arXiv:1307.5345v2 [math.GT].
  • [10] M. Gromov, “Hyperbolic groups” in Essays in Group Theory, Math. Sci. Res. Inst. Publ. 8, Springer, New York, 1987, 75–263.
  • [11] E. Guentner, “Permanence in coarse geometry” in Recent Progress in General Topology, III, Atlantis Press, Paris, 2014, 507–533.
  • [12] E. Guentner, R. Tessera, and G. Yu, A notion of geometric complexity and its application to topological rigidity, Invent. Math. 189 (2012), no. 2, 315–357.
  • [13] E. Guentner, R. Tessera, and G. Yu, Discrete groups with finite decomposition complexity, Groups Geom. Dyn. 7 (2013), no. 2, 377–402.
  • [14] R. Ji and B. Ramsey, The isocohomological property, higher Dehn functions, and relatively hyperbolic groups, Adv. Math. 222 (2009), no. 1, 255–280.
  • [15] D. Kasprowski, On the $K$-theory of groups with finite decomposition complexity, Proc. Lond. Math. Soc. (3) 110 (2015), no. 3, 565–592.
  • [16] D. Kasprowski, personal communication, August 2015.
  • [17] I. Mineyev and A. Yaman, Relative hyperbolicity and bounded cohomology, preprint, (accessed 7 January 2019).
  • [18] D. Osin, Asymptotic dimension of relatively hyperbolic groups, Int. Math. Res. Not. IMRN 2005, no. 35, 2143–2161.
  • [19] D. Osin, Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems, Mem. Amer. Math. Soc. 179 (2006), no. 843.
  • [20] N. Ozawa, Boundary amenability of relatively hyperbolic groups, Topology Appl. 153 (2006), no. 14, 2624–2630.
  • [21] D. A. Ramras, R. Tessera, and G. Yu, Finite decomposition complexity and the integral Novikov conjecture for higher algebraic $K$-theory, J. Reine Angew. Math. 694 (2014), 129–178.
  • [22] J. Roe, Lectures on Coarse Geometry, Univ. Lecture Ser. 31, Amer. Math. Soc., Providence, 2003.
  • [23] A. Sisto, “Finite decomposition complexity (is preserved by relative hyperbolicity),” Alessandro Sisto’s Math Blog (blog), October 3, 2012,
  • [24] G. Yu, The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math. 139 (2000), no. 1, 201–240.