Advertisement

Relatively hyperbolic groups with fixed peripherals

  • Matthew Cordes
  • David HumeEmail author
Article
First Online:
  • 1 Downloads

Abstract

We build quasi-isometry invariants of relatively hyperbolic groups which detect the hyperbolic parts of the group; these are variations of the stable dimension constructions previously introduced by the authors.

We prove that, given any finite collection of finitely generated groups H each of which either has finite stable dimension or is non-relatively hyperbolic, there exist infinitely many quasi-isometry types of one-ended groups which are hyperbolic relative to H.

The groups are constructed using classical small cancellation theory over free products.

This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Ali05]
    E. Alibegović, A combination theorem for relatively hyperbolic groups, Bulletin of the London Mathematical Society 37 (2005), 459–466.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [BDM09]
    J. Behrstock, C. Druţu and L. Mosher, Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity, Mathematische Annalen 344 (2009), 543–595.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [BD08]
    G. C. Bell and A. N. Dranishnikov, Asymptotic dimension, Topology and its Applications 155 (2008), 1265–1296.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [BDK04]
    G. C. Bell, A. N. Dranishnikov and J. E. Keesling, On a formula for the asymptotic dimension of free products, Fundamenta Mathematicae 183 (2004), 39–45.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [BF08]
    G. C. Bell and K. Fujiwara, The asymptotic dimension of a curve graph is finite, Journal of the London Mathematical Society 77 (2008), 33–50.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [BBF15]
    M. Bestvina, K. Bromberg and K. Fujiwara, Constructing group actions on quasitrees and applications to mapping class groups, Publications Mathématiques. Institut de Hautes Études Scientifiques 122 (2015), 1–64.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [BH78]
    A. Borel and G. Harder, Existence of discrete cocompact subgroups of reductive groups over local fields, Journal für die Reine und Angewandte Mathematik 298 (1978), 53–64.MathSciNetzbMATHGoogle Scholar
  8. [Bow12]
    B. H. Bowditch, Relatively hyperbolic groups, International Journal of Algebra and Computation 22 (2012), 1250016.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [Cha95]
    C. Champetier, Propriétés statistiques des groupes de présentation finie, Advances in Mathematics 116 (1995), 197–262.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [CH17]
    M. Cordes and D. Hume, Stability and the Morse boundary, Journal of the London Mathematical Society 95 (2017), 963–988.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [Dah03]
    F. Dahmani, Combination of convergence groups, Geometry and Topology 7 (2003), 933–963.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [DGP11]
    F. Dahmani, V. Guirardel and P. Przytycki, Random groups do not split, Mathematische Annalen 349 (2011), 657–673.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [DS05]
    C. Druţu and M. Sapir, Tree-graded spaces and asymptotic cones of groups, Topology 44 (2005), 959–1058.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [Dru09]
    C. Druţu, Relatively hyperbolic groups: geometry and quasi-isometric invariance, Commentarii Mathematici Helvetici 84 (2009), 503–546.MathSciNetzbMATHGoogle Scholar
  15. [Dun85]
    M. J. Dunwoody, The accessibility of finitely presented groups, Inventiones Mathematicae 81 (1985), 449–458.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [DT15]
    M. G. Durham and S. J. Taylor, Convex cocompactness and stability in mapping class groups, Algebraic & Geometric Topology 15 (2015), 2839–2859.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [Far98]
    B. Farb, Relatively hyperbolic groups, Geometric and Functional Analysis 8 (1998), 810–840.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [Gro13]
    B. W. Groff, Quasi-isometries, boundaries and JSJ-decompositions of relatively hyperbolic groups, Journal of Topology and Analysis 5 (2013), 451–475.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [Gro87]
    M. Gromov, Hyperbolic groups, in Essays in Group Theory, Mathematical Sciences Research Institute Publications, Vol. 8, Springer, New York, 1987, pp. 75–263.Google Scholar
  20. [Gru]
    D. Gruber, Personal communication.Google Scholar
  21. [Hum17]
    D. Hume, Embedding mapping class groups into a finite product of trees, Groups, Geometry, and Dynamics 11 (2017), 613–647.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [KM10]
    O. Kharlampovich and A. G. Myasnikov, Equations and fully residually free groups, in Combinatorial and Geometric Group Theory, Trens in Mathematics, Birkhäuser/Springer, Basel, 2010, pp. 203–242.Google Scholar
  23. [LS10]
    R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Classics in Mathematics, Springer-Verlag, Berlin, 2001.CrossRefzbMATHGoogle Scholar
  24. [Mac12]
    J. M. Mackay, Conformal dimension and random groups, Geometric and Functional Analysis 22 (2012), 213–239.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [MS13]
    J. M. Mackay and A. Sisto, Embedding relatively hyperbolic groups in products of trees, Algebraic & Geometric Topology 13 (2013), 2261–2282.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [Osi05]
    D. V. Osin, Asymptotic dimension of relatively hyperbolic groups, International Mathematics Research Notices 35 (2005), 2143–2161.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [Osi06]
    D. V. Osin, Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems, Memoirs of the American Mathematical Society 179 (2006).Google Scholar
  28. [PW02]
    P. Papasoglu and K. Whyte, Quasi-isometries between groups with infinitely many ends, Commentarii Mathematici Helvetici 77 (2002), 133–144.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [Roe05]
    J. Roe, Hyperbolic groups have finite asymptotic dimension, Proceedings of the American Mathematical Society 133 (2005), 2489–2490.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [Sch95]
    R. Schwartz, The quasi-isometry classification of rank one lattices, Publications Mathématiques. Institut des Hautes Etudes Scientifiques 82 (1995), 133–168.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [Sel01]
    Z. Sela, Diophantine geometry over groups. I. Makanin–Razborov diagrams, Publications Mathématiques. Institut des Hautes Études Scientifiques 93 (2001), 31–105.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [Sil03]
    L. Silberman, Addendum to: “Random walk in random groups” [Geom. Funct. Anal. 13 (2003), no. 1, 73–146; MR1978492] by M. Gromov, Geometric and Functional Analysis 13 (2003), 147–177.MathSciNetCrossRefGoogle Scholar
  33. [Sis13]
    A. Sisto, Projections and relative hyperbolicity, L’Enseignement Mathématique 59 (2013), 165–181.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [Sta68]
    J. Stallings, Groups of dimension 1 are locally free, Bulletin of the American Mathematical Society 74 (1968), 361–364.MathSciNetCrossRefzbMATHGoogle Scholar
  35. [Sta71]
    J. Stallings, Group Theory and Three-Dimensional Manifolds, Yale Mathematical Monographs, Vol. 4, Yale University Press, New Haven, CT, 1971.Google Scholar
  36. [Str90]
    R. Strebel, Appendix. Small cancellation groups, in Sur les groupes hyperboliques d’après Mikhael Gromov (Bern, 1988), Progress in Mathematics, Vol. 83, Birkhäuser Boston, Boston, MA, 1990, pp. 227–273.MathSciNetGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2019

Authors and Affiliations

  1. 1.Department of MathematicsETH ZurichZurichSwitzerland
  2. 2.Matheamtical InstituteUniversity of OxfordOxfordUK

Personalised recommendations

Cite article

Buy options