Geometriae Dedicata

, Volume 180, Issue 1, pp 339–371 | Cite as

A Cartan–Hadamard type result for relatively hyperbolic groups

Original Paper Series A


In this article, we prove that if a finitely presented group has an asymptotic cone which is tree-graded with respect to a precise set of pieces then it is relatively hyperbolic. This answers a question of Mark Sapir and generalizes a result of Kapovich and Kleiner to relatively hyperbolic groups.


Relatively hyperbolic Asymptotic cones Tree-graded spaces 

Mathematics Subject Classification

20F65 20F67 20F69 


  1. 1.
    Anderson, J.W., Aramayona, J., Shackleton, K.J.: An obstruction to the strong relative hyperbolicity of a group. J. Group Theory 10(6), 749–756 (2007)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Arzhantseva, GN., Delzant, T.: Examples of random groups (Nov 2008).
  3. 3.
    Behrstock, J.A.: Asymptotic geometry of the mapping class group and Teichmüller space. Geom. Topol. 10, 1523–1578 (2006)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Behrstock, J.A., Druţu, C., Mosher, L.: Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity. Math. Ann. 344(3), 543–595 (2009)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Bourbaki, N.: Éléments de mathématique. Topologie générale. Chapitres 1 à 4. Hermann, Paris (1971)Google Scholar
  6. 6.
    Bowditch, B.H.: Notes on Gromov’s hyperbolicity criterion for path-metric spaces. In: Group theory from a Geometrical Viewpoint (Trieste, 1990), pp 64–167. World Sci. Publ., River Edge, NJ, (1991)Google Scholar
  7. 7.
    Bowditch, B.H.: Relatively hyperbolic groups. Int. J. Algebra Comput. 22(3), 1250016–1250066 (2012)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature Volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1999)Google Scholar
  9. 9.
    Coornaert, M., Delzant, T., Papadopoulos, A.: Géométrie et théorie des groupes, volume 1441 of Lecture Notes in Mathematics. Springer, Berlin (1990)Google Scholar
  10. 10.
    Coulon, R.: Asphericity and small cancellation theory for rotation families of groups. Groups Geom. Dyn. 5(4), 729–765 (2011)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Coulon, R.: On the geometry of Burnside quotients of torsion free hyperbolic groups. Int. J. Algebra Comput. 24(3), 251–345 (2014)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Delzant, T., Gromov, M.: Courbure mésoscopique et théorie de la toute petite simplification. J. Topol. 1(4), 804–836 (2008)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Druţu, C.: Quasi-isometry invariants and asymptotic cones. Int. J. Algebra Comput. 12(1–2), 99–135 (2002)MATHGoogle Scholar
  14. 14.
    Druţu, C.: Relatively hyperbolic groups: geometry and quasi-isometric invariance. Comment. Math. Helv. 84(3), 503–546 (2009)MathSciNetMATHGoogle Scholar
  15. 15.
    Druţu, C., Sapir, M.V.: Tree-graded spaces and asymptotic cones of groups. Topol. Int. J. Math. 44(5), 959–1058 (2005)MATHGoogle Scholar
  16. 16.
    Druţu, C., Sapir, M.V.: Groups acting on tree-graded spaces and splittings of relatively hyperbolic groups. Adv. Math. 217(3), 1313–1367 (2008)CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Farb, B.: Relatively hyperbolic groups. Geom. Funct. Anal. 8(5), 810–840 (1998)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Gromov, M.: Groups of polynomial growth and expanding maps. Publ. Math. Inst. Ht. Études Sci. 53, 53–73 (1981)CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Gromov, M.: Hyperbolic groups. In: Gersten, S.M. (ed.) Essays in group theory, pp. 75–263. Springer, New York (1987)Google Scholar
  20. 20.
    Gromov, M.: Asymptotic invariants of infinite groups. In Geometric group theory, vol. 2 (Sussex, 1991), pp 1–295. Cambridge University Press, Cambridge (1993)Google Scholar
  21. 21.
    Gromov, M.: Random walk in random groups. Geom. Funct. Anal. 13(1), 73–146 (2003)CrossRefMathSciNetMATHGoogle Scholar
  22. 22.
    Groves, D.: Limit groups for relatively hyperbolic groups. II. Makanin–Razborov diagrams. Geom. Topol. 9, 2319–2358 (2005)CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    Groves, D.: Limit groups for relatively hyperbolic groups. I. The basic tools. Algebr. Geom. Topol. 9(3), 1423–1466 (2009)CrossRefMathSciNetMATHGoogle Scholar
  24. 24.
    Groves, D., Manning, J.: Dehn filling in relatively hyperbolic groups. Isr. J. Math. 168, 317–429 (2008)CrossRefMathSciNetMATHGoogle Scholar
  25. 25.
    Guirardel, V.: Actions of finitely generated groups on \(\mathbb{R}\)-trees. Ann. Inst. Fourier 58(1), 159–211 (2008). (Grenoble)CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    Hruska, G.C., Kleiner, B.: Hadamard spaces with isolated flats. Geom. Topol., 9:1501–1538, (2005). With an appendix by the authors and Mohamad HindawiGoogle Scholar
  27. 27.
    Lyndon, R.C., Schupp, P.E., Schupp, P.E.: Comb. group theory. Springer, Berlin (1977)Google Scholar
  28. 28.
    Minsky, Y.N., Behrstock, J.A., Kleiner, B., Mosher, L.: Geometry and rigidity of mapping class groups. Geom. Topol. 16(2), 781–888 (2012)CrossRefMathSciNetMATHGoogle Scholar
  29. 29.
    Ol’shanskiĭ, A.Y.: An infinite group with subgroups of prime orders. Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, 44(2):309–321, 479 (1980)Google Scholar
  30. 30.
    Ol’shanskiĭ, A.Y., Sapir, M.V., Osin, D.V.: Lacunary hyperbolic groups. Geom. Topol. 13(4), 2051–2140 (2009)CrossRefMathSciNetMATHGoogle Scholar
  31. 31.
    Osin, D., Sapir, M.: Universal tree-graded spaces and asymptotic cones. Int. J. Algebra Comput. 21(5), 793–824 (2011)CrossRefMathSciNetMATHGoogle Scholar
  32. 32.
    Osin, D.V.: Elementary subgroups of relatively hyperbolic groups and bounded generation. Int. J. Algebra Comput. 16(1), 99–118 (2006)CrossRefMathSciNetMATHGoogle Scholar
  33. 33.
    Osin, D.V.: Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems. Memoirs of the American Mathematical Society, 179(843):vi–100 (2006)Google Scholar
  34. 34.
    Osin, D.V.: Small cancellations over relatively hyperbolic groups and embedding theorems. Ann. Math. Second Ser. 172(1), 1–39 (2010)CrossRefMathSciNetMATHGoogle Scholar
  35. 35.
    Osin, D.: Acylindrically hyperbolic groups. Trans. Am. Math. Soc. (2015). doi:10.1090/tran/6343
  36. 36.
    Papasoglu, P.: An algorithm detecting hyperbolicity. In: Geometric and Computational Perspectives on Infinite Groups (Minneapolis. MN and New Brunswick, NJ, 1994), pp. 193–200. American Mathematical Society, Providence, RI (1996)Google Scholar
  37. 37.
    Sisto, A.: Tree-graded asymptotic cones. Groups Geom. Dyn. 7(3), 697–735 (2013)CrossRefMathSciNetMATHGoogle Scholar
  38. 38.
    Stallings, J.: On torsion-free groups with infinitely many ends. Ann. Math. Second Ser. 88, 312–334 (1968)CrossRefMathSciNetMATHGoogle Scholar
  39. 39.
    Stallings, J.: Group Theory and Three-Dimensional Manifolds. Yale University Press, New Haven (1971)MATHGoogle Scholar
  40. 40.
    van den Dries, L., Wilkie, A.J.: Gromov’s theorem on groups of polynomial growth and elementary logic. J. Algebra 89(2), 349–374 (1984)CrossRefMathSciNetMATHGoogle Scholar
  41. 41.
    Yaman, A.: A topological characterisation of relatively hyperbolic groups. Journal für die Reine und Angewandte Mathematik. [Crelle’s Journal], 566:41–89 (2004)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.CNRS/IRMAR, Université de Rennes 1Rennes CedexFrance
  2. 2.Department of Mathematics, Statistics, and Computer ScienceUniversity of Illinois at ChicagoChicagoUSA
  3. 3.Courant Institute of MathematicsNew York UniversityNew YorkUSA

Personalised recommendations