Geometriae Dedicata

, Volume 169, Issue 1, pp 209–224 | Cite as

Hyperbolic quasi-geodesics in CAT(0) spaces

  • Harold Sultan
Original paper


We prove the equivalence of various hyperbolic type properties for quasi-geodesics in CAT(0) spaces. Specifically, in our main theorem we prove that for \(X\) a CAT(0) space and \(\gamma \subset X\) a quasi-geodesic, the following four statements are equivalent: (1) \(\gamma \) is Morse, (2) \(\gamma \) is (b,c)—contracting, (3) \(\gamma \) is strongly contracting, and (4) in every asymptotic cone \(X_{\omega },\) any two distinct points in the ultralimit \(\gamma _{\omega }\) are separated by a cut-point. In particular, this characterization provides a converse to a generalized Morse stability lemma in the CAT(0) setting. In addition, under mild conditions we prove the equivalence of wideness and unconstrictedness in the CAT(0) setting.


Hyperbolic type geodesics Contracting geodesics Asymptotic cone 

Mathematics Subject Classification (2010)

20F65 20F67 



I want to express my gratitude to my advisors Jason Behrstock and Walter Neumann for their extremely helpful advice and insights throughout my research, and specifically regarding this paper. I would also like to acknowledge Igor Belegradek, Ruth Charney, and Amey Kaloti for useful conversations and insights regarding arguments and ideas in this paper.


  1. 1.
    Algom-Kfir, Y.: Strongly contracting geodesics in outer space. Preprint arXiv:0812.155 (2010)Google Scholar
  2. 2.
    Behrstock, J.: Asymptotic geometry of the mapping class group and Teichmüller space. Geom. Topol. 10, 1523–1578 (2006)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Behrstock, J., Charney, R.: Divergence and quasimorphisms of right-angled Artin groups. Math. Ann. 1–18 (2011)Google Scholar
  4. 4.
    Behrstock, J., Drutu, C.: Divergence, thick groups, and short conjugators, Preprint 1110.5005 (2011)Google Scholar
  5. 5.
    Bestvina, M., Fujiwara, K.: A characterization of higher rank symmetric spaces via bounded cohomology. Geom. Funct. Anal. 19, 11–40 (2009)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Bridson, M., Haefliger, A.: Metric spaces of non-Positive curvature, Grad. Texts in Math. 319. Springer, New York (1999)CrossRefGoogle Scholar
  7. 7.
    Brock, J., Farb, B.: Curvature and rank of Teichmüller space. Am. J. Math. 128, 1–22 (2006)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Brock, J., Masur, H.: Coarse and synthetic Weil-Petersson geometry: quasiats, geodesics, and relative hyperbolicity. Geom. Topol. 12, 2453–2495 (2008)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Brock, J., Masur, H., Minsky, Y.: Asymptotics of Weil-Petersson geodesics II: bounded geometry and unbounded entropy. Geom. Funct. Anal. 21, 820–850 (2011)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Charney, R.: Lecture entitled “Contracting boundaries of CAT(0) spaces”, at 3-manifolds, Artin Groups, and cubical geometry CBMS-NSF conference. CUNY, New York (2011)Google Scholar
  11. 11.
    Drutu, C., Mozes, S., Sapir, M.: Divergence in lattices in semisimple Lie groups and graphs of groups. Trans. Amer. Math. Soc. 362, 2451–2505 (2010)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Drutu, C., Sapir, M.: Tree-graded spaces and asymptotic cones of groups. Topology 44, 959–1058 (2005). With an appendix by Denis Osin and Mark SapirGoogle Scholar
  13. 13.
    Kapovich, M., Leeb, B.: 3-Manifold groups and nonpositive curvature. Geom. Funct. Anal. 8, 841–852 (1998)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Masur, H., Minsky, Y.: Geometry of the complex of curves. I. Hyperbolicity. Invent. Math. 138, 103–149 (1999)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Morse, H.: A fundamental class of geodesics on any closed surface of genus greater than one. Trans. AMS 26, 25–60 (1924)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Mosher, L.: Stable Teichmüller quasigeodesics and ending laminations. Geom. Topol. 7, 33–90 (2003)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Osin, D.: Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems. Mem. Am. Math. Soc. 179(843), vi+100pp (2006)Google Scholar
  18. 18.
    Sultan, H.: The asymptotic cone of Teichmüller space: thickness and divergence. Ph.D. thesis, Columbia University (2012)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of MathematicsBrandeis UniversityWalthamUSA

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