Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The Tits boundary of a $\text{CAT}(0)$ 2-complex

Author: Xiangdong Xie
Journal: Trans. Amer. Math. Soc. 357 (2005), 1627-1661
MSC (2000): Primary 20F67, 20F65; Secondary 57M20, 53C20
Published electronically: October 28, 2004
MathSciNet review: 2115379
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the Tits boundary of $\text{CAT}(0)$ $2$-complexes that have only a finite number of isometry types of cells. In particular, we show that away from the endpoints, a geodesic segment in the Tits boundary is the ideal boundary of an isometrically embedded Euclidean sector. As applications, we provide sufficient conditions for two points in the Tits boundary to be the endpoints of a geodesic in the $2$-complex and for a group generated by two hyperbolic isometries to contain a free group. We also show that if two $\text{CAT}(0)$ $2$-complexes are quasi-isometric, then the cores of their Tits boundaries are bi-Lipschitz.

References [Enhancements On Off] (What's this?)

  • [B] W. Ballmann, Lectures on spaces of nonpositive curvature, volume 25 of DMV seminar. Birkhäuser, 1995. MR 97a:53053
  • [BBr] W. Ballmann, M. Brin, Orbihedra of nonpositive curvature, Inst. Hautes Études Sci. Publ. Math. 82 (1996), 169-209.MR 97i:53049
  • [BGS] W. Ballmann, M. Gromov and V. Schroeder, Manifolds of Nonpositive Curvature, Birkhäuser, 1985. MR 87h:53050
  • [BH] M. Bridson, A. Haefliger, Metric spaces of nonpositive curvature, Grundlehren 319, Springer-Verlag, Berlin(1999).MR 2000k:53038
  • [Br] M. Bridson, On the semisimplicity of polyhedral isometries, Proceedings of the American Mathematical Society 127, no. 7, 2143-2146.MR 99m:53086
  • [BS] S. Buyalo, V. Schroeder, On the asymptotic geometry of nonpositively curved graph manifolds, Trans. Amer. Math. Soc. 353, no.3 (2001), 853-875.MR 2001f:53063
  • [CK1] C. Croke, B. Kleiner, Spaces with nonpositive curvature and their ideal boundaries, Topology 39 (2000) 549-556. MR 2001b:53035
  • [CK2] C. Croke, B. Kleiner, The geodesic flow of a nonpositively curved graph manifold, Geom. Funct. Anal. 12 (2002), no. 3, 479-545.MR 2003i:53057
  • [H] J. Heber, On the geometric rank of homogeneous spaces of nonpositive curvature, Invent. Math. 112 (1993), no. 1, 151-170. MR 94a:53082
  • [HS1] C. Hummel, V. Schroeder, Tits geometry associated with 4-dimensional closed real-analytic manifolds of nonpositive curvature, J. Differential Geometry 48 (1998), 531-555. MR 99j:53058
  • [HS2] C. Hummel, V. Schroeder, Tits geometry of cocompact real-analytic Hadamard manifolds of dimension 4, Differential Geometry and its Applications 11 (1999), 129-143. MR 2000h:53052
  • [K] B. Kleiner, The local structure of length spaces with curvature bounded above, Math. Z. 231 no. 3 (1999), 409-456. MR 2000m:53053
  • [K2] B. Kleiner, private notes.
  • [KO] K. Kawamura, F. Ohtsuka, Total excess and Tits metric for piecewise Riemannian 2-manifolds, Topology and its Applications 94 (1999) 173-193.MR 2000d:53066
  • [L] B. Leeb, A characterization of irreducible symmetric spaces and Euclidean buildings of higher rank by their asymptotic geometry, Bonn Mathematical Publications 326. MR 2004b:53060
  • [M] G. Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies, No. 78, Princeton University Press, 1973.MR 52:5874
  • [N] I. Nikolaev, The tangent cone of an Aleksandrov space of curvature $\le K$, Manuscripta Math. 86 no. 2 (1995), 137-147. MR 95m:53062
  • [R] K. Ruane, Dynamics of the Action of a $\operatorname{CAT}(0)$ group on the Boundary, Geometriae Dedicata 84 (2001), 81-99. MR 2002d:20064
  • [X1] X. Xie, Tits alternative for closed real analytic 4-manifolds of nonpositive curvature, Topology and its applications 136 (2004), 87-121.
  • [X2] X. Xie, Groups Acting on $\operatorname{CAT}(0)$ Square Complexes, to appear in Geometriae Dedicata.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 20F67, 20F65, 57M20, 53C20

Retrieve articles in all journals with MSC (2000): 20F67, 20F65, 57M20, 53C20

Additional Information

Xiangdong Xie
Affiliation: Department of Mathematics, Washington University, St. Louis, Missouri 63130
Address at time of publication: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221

Keywords: Tits boundary, Tits metric, $\text{CAT}(0)$, 2-complex, quasi-isometry, quasi-flat
Received by editor(s): March 10, 2003
Received by editor(s) in revised form: December 1, 2003
Published electronically: October 28, 2004
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society