Geometric & Functional Analysis GAFA

, Volume 4, Issue 1, pp 37–51 | Cite as

Quadratic divergence of geodesics in CAT(0) spaces

  • S. M. Gersten


A finite CAT(0) 2-complexX is produced whose universal cover possesses two geodesic rays which diverge quadratically and such that no pair of rays diverges faster than quadratically. This example shows that an aphorism in Riemannian geometry, that predicts that in nonpositive curvature nonasymptotic geodesic rays either diverge exponentially or diverge linearly, does not hold in the setting of CAT(0) complexes. The fundamental group ofX is that of a compact Riemannian manifold with totally geodesic boundary and nonpositive sectional curvature.


Riemannian Manifold Fundamental Group Sectional Curvature Universal Cover Riemannian Geometry 
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  1. [A]J. Alonso, Inégalités isopérimétriques et quasi-isométries, C.R. Acad. Sci. Paris Série 1 311 (1990), 761–764.Google Scholar
  2. [BGrS]W. Ballmann, M. Gromov, V. Schroeder, Manifolds of Nonpositive Curvature, Birkhäuser, Progress in Math. 61 (1985).Google Scholar
  3. [BeFe]M. Bestvina, M. Feighn, A combination theorem for negatively curved groups Jour. Diff. Geom. 35 (1992), 85–101.Google Scholar
  4. [Br]M.R. Bridson, Geodesics and curvature in metric simplicial complexes, in “Group Theory from a Geometrical Viewpoint” (E. Ghys. A. Haefliger, A. Verjovsky, eds.), World Scientific 1991.Google Scholar
  5. [BrG]M.R. Bridson, S. M. Gersten, The optimal isoperimetric inequality for ℤnAℤ, preprint 1992, Univ. of Utah.Google Scholar
  6. [F]B. Farb, The extrinsic geometry of subgroups and the generalized word problem, to appear in Proc. London Math. Soc.Google Scholar
  7. [Fl]W. Floyd, Group completions and limit sets of Kleinian groups,Inv. Math. 56 (1980), 205–218.Google Scholar
  8. [Fr]D. Fried, Monodromy and dynamical systems, Topology 25 (1986), 443–453.Google Scholar
  9. [G1]S.M. Gersten, The automorphism group of a free group is not a CAT(0) group, to appear in Proc. Amer. Math. Soc.Google Scholar
  10. [G2]S.M. Gersten, Isoperimetric and isodiametric inequalities of finite presentations, in “Geometric Group Theory, Volume I” (G. Niblo, M. Roller, eds.), London Math. Soc. Lecture Notes Series 181 (1993), 79–96, Cambridge Univ. Press.Google Scholar
  11. [GhH]E. Ghys, P. de la Harpe, Sur les Groupes Hyperboliques d'après Mikhael Gromov, Progress in Math. 83, Birkhäuser, 1990.Google Scholar
  12. [Gr1]M. Gromov, Hyperbolic groups, “Essays in Group Theory” (S.M. Gersten, ed.), Springer-Verlag, MSRI series 8, 1987.Google Scholar
  13. [Gr2]M. Gromov, Asymptotic Invariants of Infinite Groups, “Geometric Group Theory, Volume 2” (G. Niblo, M. Roller, eds.), London Math. Soc. Lecture Notes Series 182 (1993), Cambridge Univ. Press,Google Scholar
  14. [GrLP]M. Gromov, J. Lafontaine, P. Pansu, Structures Métriques pour les Variétés Riemanniennes, Cedic/Fernand Nathan, 1981.Google Scholar
  15. [GrP]M. Gromov and P. Pansu, Rigidity of lattices: an introduction, preprint IHES.Google Scholar
  16. [GSh]S.M. Gersten, H. Short, Small cancellation theory and automatic groups, Inv. Math. 102 (1990), 305–334.Google Scholar
  17. [KLe]M. Kapovich, B. Leeb, in preparation.Google Scholar
  18. [Le]B. Leeb, Metrics of nonpositive curvature on 3-manifolds, Ph.D. Thesis, Univ. of Maryland, 1993.Google Scholar
  19. [LySc]R.C. Lyndon, P.E. Schupp, Combinatorial Group Theory, Springer-Verlag, 1977.Google Scholar
  20. [P]P. Pansu, Croissance des boules et des géodésiques fermées dans les nilvariétés, Ergod. Th. Dynam. Syst. 3 (1983), 415–445.Google Scholar
  21. [Sh]H. Short, ed., Notes on word hyperbolic groups, in “Group Theory from a Geometrical Viewpoint” (E. Ghys, A. Haefliger, A. Verjovsky, eds.), World Scientific 1991.Google Scholar
  22. [St]J. Stallings, Topologically unrealizable automorphisms of free groups, Proc. Amer. Math. Soc. 84 (1982), 21–24.Google Scholar

Copyright information

© Birkhäuser Verlag 1994

Authors and Affiliations

  • S. M. Gersten
    • 1
  1. 1.Mathematics Dept.University of UtahSalt Lake CityUSA

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