The Tits boundary of a 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 Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the Tits boundary of -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 -complex and for a group generated by two hyperbolic isometries to contain a free group. We also show that if two -complexes are quasi-isometric, then the cores of their Tits boundaries are bi-Lipschitz.

**[B]**Werner Ballmann,*Lectures on spaces of nonpositive curvature*, DMV Seminar, vol. 25, Birkhäuser Verlag, Basel, 1995. With an appendix by Misha Brin. MR**1377265****[BBr]**Werner Ballmann and Michael Brin,*Orbihedra of nonpositive curvature*, Inst. Hautes Études Sci. Publ. Math.**82**(1995), 169–209 (1996). MR**1383216****[BGS]**Werner Ballmann, Mikhael Gromov, and Viktor Schroeder,*Manifolds of nonpositive curvature*, Progress in Mathematics, vol. 61, Birkhäuser Boston, Inc., Boston, MA, 1985. MR**823981****[BH]**Martin R. Bridson and André Haefliger,*Metric spaces of non-positive curvature*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR**1744486****[Br]**Martin R. Bridson,*On the semisimplicity of polyhedral isometries*, Proc. Amer. Math. Soc.**127**(1999), no. 7, 2143–2146. MR**1646316**, 10.1090/S0002-9939-99-05187-4**[BS]**S. Buyalo and V. Schroeder,*On the asymptotic geometry of nonpositively curved graphmanifolds*, Trans. Amer. Math. Soc.**353**(2001), no. 3, 853–875. MR**1707192**, 10.1090/S0002-9947-00-02583-6**[CK1]**Christopher B. Croke and Bruce Kleiner,*Spaces with nonpositive curvature and their ideal boundaries*, Topology**39**(2000), no. 3, 549–556. MR**1746908**, 10.1016/S0040-9383(99)00016-6**[CK2]**C. B. Croke and B. Kleiner,*The geodesic flow of a nonpositively curved graph manifold*, Geom. Funct. Anal.**12**(2002), no. 3, 479–545. MR**1924370**, 10.1007/s00039-002-8255-7**[H]**Jens Heber,*On the geometric rank of homogeneous spaces of nonpositive curvature*, Invent. Math.**112**(1993), no. 1, 151–170. MR**1207480**, 10.1007/BF01232428**[HS1]**Christoph Hummel and Viktor Schroeder,*Tits geometry associated with 4-dimensional closed real-analytic manifolds of nonpositive curvature*, J. Differential Geom.**48**(1998), no. 3, 531–555. MR**1638057****[HS2]**Christoph Hummel and Viktor Schroeder,*Tits geometry of cocompact real-analytic Hadamard manifolds of dimension 4*, Differential Geom. Appl.**11**(1999), no. 2, 129–143. MR**1712131**, 10.1016/S0926-2245(99)00025-X**[K]**Bruce Kleiner,*The local structure of length spaces with curvature bounded above*, Math. Z.**231**(1999), no. 3, 409–456. MR**1704987**, 10.1007/PL00004738**[K2]**B. Kleiner,*private notes.***[KO]**Kazuhiro Kawamura and Fumiko Ohtsuka,*Total excess and Tits metric for piecewise Riemannian 2-manifolds*, Topology Appl.**94**(1999), no. 1-3, 173–193. Special issue in memory of B. J. Ball. MR**1695355**, 10.1016/S0166-8641(98)00077-7**[L]**Bernhard Leeb,*A characterization of irreducible symmetric spaces and Euclidean buildings of higher rank by their asymptotic geometry*, Bonner Mathematische Schriften [Bonn Mathematical Publications], 326, Universität Bonn, Mathematisches Institut, Bonn, 2000. MR**1934160****[M]**G. D. Mostow,*Strong rigidity of locally symmetric spaces*, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973. Annals of Mathematics Studies, No. 78. MR**0385004****[N]**Igor Nikolaev,*The tangent cone of an Aleksandrov space of curvature ≤𝐾*, Manuscripta Math.**86**(1995), no. 2, 137–147. MR**1317739**, 10.1007/BF02567983**[R]**Kim E. Ruane,*Dynamics of the action of a 𝐶𝐴𝑇(0) group on the boundary*, Geom. Dedicata**84**(2001), no. 1-3, 81–99. MR**1825346**, 10.1023/A:1010301824765**[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 Square Complexes,*to appear in Geometriae Dedicata.

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

Email:
xxie@math.wustl.edu, xiexg@ucmail.uc.edu

DOI:
https://doi.org/10.1090/S0002-9947-04-03575-5

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