Tokyo Journal of Mathematics

On Relative Hyperbolicity for a Group and Relative Quasiconvexity for a Subgroup

Yoshifumi MATSUDA, Shin-ichi OGUNI, and Saeko YAMAGATA

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


We consider two families of subgroups of a group. Assume that each subgroup which belongs to one family is contained in some subgroup which belongs to the other family. We then discuss relations of relative hyperbolicity for the group with respect to the two families, respectively. If the group is hyperbolic relative to the two families, respectively, then we consider relations of relative quasiconvexity for a subgroup of the group with respect to the two families, respectively.

Article information

Tokyo J. Math., Volume 42, Number 1 (2019), 83-112.

First available in Project Euclid: 18 July 2019

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F67: Hyperbolic groups and nonpositively curved groups
Secondary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]


MATSUDA, Yoshifumi; OGUNI, Shin-ichi; YAMAGATA, Saeko. On Relative Hyperbolicity for a Group and Relative Quasiconvexity for a Subgroup. Tokyo J. Math. 42 (2019), no. 1, 83--112.

Export citation


  • \BibAuthorsB. H. Bowditch, Relatively hyperbolic groups, Internat. J. Algebra. Comput. 22 (2012), 1250016, 66 pp.
  • \BibAuthorsF. Dahmani, V. Guirardel and D. Osin, Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces, Mem. Amer. Math. Soc. 245 (2017), no. 1156, v+152 pp.
  • \BibAuthorsC. Druţu and M. Sapir, Tree-graded spaces and asymptotic cones of groups. With an appendix by Denis Osin and Mark Sapir, Topology 44 (2005), 959–1058.
  • \BibAuthorsB. Farb, Relatively hyperbolic groups, Geom. Funct. Anal. 8 (1998), 810–840.
  • \BibAuthorsM. Gromov, Hyperbolic groups, in Essays in Group Theory, 75–263, Math. Sci. Res. Inst. Publ. 8, Springer, New York, 1987.
  • \BibAuthorsG. C. Hruska, Relative hyperbolicity and relative quasiconvexity for countable groups, Algebr. Geom. Topol. 10 (2010), 1807–1856.
  • \BibAuthorsE. Martínez-Pedroza, On quasiconvexity and relative hyperbolic structures on groups, Geom. Dedicata 157 (2012), 269–290.
  • \BibAuthorsY. Matsuda, S. Oguni and S. Yamagata, The universal relatively hyperbolic structure on a group and relative quasiconvexity for subgroups, Analysis and Geometry of Discrete Groups and Hyperbolic Spaces, 73–93, RIMS Kôkyûroku Bessatsu, B48, Res. Inst. Math. Sci. (RIMS), Kyoto, 2014.
  • \BibAuthorsY. Matsuda, S. Oguni and S. Yamagata, Notes on relatively hyperbolic groups and relatively quasiconvex subgroups, Tokyo J. Math. 38 (2015), 99–123.
  • \BibAuthorsD. V. Osin, Relatively hyperbolic groups: Intrinsic geometry, algebraic properties, and algorithmic problems, Mem. Amer. Math. Soc. 179 (2006), no. 843, vi+100 pp.
  • \BibAuthorsD. V. Osin, Elementary subgroups of relatively hyperbolic groups and bounded generation, Internat. J. Algebra. Comput. 16 (2006), 99–118.
  • \BibAuthorsJ.-P. Serre, Trees, Springer Monogr. Math., Springer-Verlag, Berlin, 2003.
  • \BibAuthorsW. Yang, Peripheral structures of relatively hyperbolic groups, J. Reine Angew. Math. 689 (2014), 101–135.