The Tits boundary of a 2complex
Author:
Xiangdong Xie
Journal:
Trans. Amer. Math. Soc. 357 (2005), 16271661
MSC (2000):
Primary 20F67, 20F65; Secondary 57M20, 53C20
Published electronically:
October 28, 2004
MathSciNet review:
2115379
Fulltext 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 quasiisometric, then the cores of their Tits boundaries are biLipschitz.
 [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), 169209.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, SpringerVerlag, Berlin(1999).MR 2000k:53038
 [Br]
M. Bridson, On the semisimplicity of polyhedral isometries, Proceedings of the American Mathematical Society 127, no. 7, 21432146.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), 853875.MR 2001f:53063
 [CK1]
C. Croke, B. Kleiner, Spaces with nonpositive curvature and their ideal boundaries, Topology 39 (2000) 549556. MR 2001b:53035
 [CK2]
C. Croke, B. Kleiner, The geodesic flow of a nonpositively curved graph manifold, Geom. Funct. Anal. 12 (2002), no. 3, 479545.MR 2003i:53057
 [H]
J. Heber, On the geometric rank of homogeneous spaces of nonpositive curvature, Invent. Math. 112 (1993), no. 1, 151170. MR 94a:53082
 [HS1]
C. Hummel, V. Schroeder, Tits geometry associated with 4dimensional closed realanalytic manifolds of nonpositive curvature, J. Differential Geometry 48 (1998), 531555. MR 99j:53058
 [HS2]
C. Hummel, V. Schroeder, Tits geometry of cocompact realanalytic Hadamard manifolds of dimension 4, Differential Geometry and its Applications 11 (1999), 129143. MR 2000h:53052
 [K]
B. Kleiner, The local structure of length spaces with curvature bounded above, Math. Z. 231 no. 3 (1999), 409456. MR 2000m:53053
 [K2]
B. Kleiner, private notes.
 [KO]
K. Kawamura, F. Ohtsuka, Total excess and Tits metric for piecewise Riemannian 2manifolds, Topology and its Applications 94 (1999) 173193.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 , Manuscripta Math. 86 no. 2 (1995), 137147. MR 95m:53062
 [R]
K. Ruane, Dynamics of the Action of a group on the Boundary, Geometriae Dedicata 84 (2001), 8199. MR 2002d:20064
 [X1]
X. Xie, Tits alternative for closed real analytic 4manifolds of nonpositive curvature, Topology and its applications 136 (2004), 87121.
 [X2]
X. Xie, Groups Acting on Square Complexes, to appear in Geometriae Dedicata.
 [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), 169209.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, SpringerVerlag, Berlin(1999).MR 2000k:53038
 [Br]
 M. Bridson, On the semisimplicity of polyhedral isometries, Proceedings of the American Mathematical Society 127, no. 7, 21432146.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), 853875.MR 2001f:53063
 [CK1]
 C. Croke, B. Kleiner, Spaces with nonpositive curvature and their ideal boundaries, Topology 39 (2000) 549556. MR 2001b:53035
 [CK2]
 C. Croke, B. Kleiner, The geodesic flow of a nonpositively curved graph manifold, Geom. Funct. Anal. 12 (2002), no. 3, 479545.MR 2003i:53057
 [H]
 J. Heber, On the geometric rank of homogeneous spaces of nonpositive curvature, Invent. Math. 112 (1993), no. 1, 151170. MR 94a:53082
 [HS1]
 C. Hummel, V. Schroeder, Tits geometry associated with 4dimensional closed realanalytic manifolds of nonpositive curvature, J. Differential Geometry 48 (1998), 531555. MR 99j:53058
 [HS2]
 C. Hummel, V. Schroeder, Tits geometry of cocompact realanalytic Hadamard manifolds of dimension 4, Differential Geometry and its Applications 11 (1999), 129143. MR 2000h:53052
 [K]
 B. Kleiner, The local structure of length spaces with curvature bounded above, Math. Z. 231 no. 3 (1999), 409456. MR 2000m:53053
 [K2]
 B. Kleiner, private notes.
 [KO]
 K. Kawamura, F. Ohtsuka, Total excess and Tits metric for piecewise Riemannian 2manifolds, Topology and its Applications 94 (1999) 173193.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 , Manuscripta Math. 86 no. 2 (1995), 137147. MR 95m:53062
 [R]
 K. Ruane, Dynamics of the Action of a group on the Boundary, Geometriae Dedicata 84 (2001), 8199. MR 2002d:20064
 [X1]
 X. Xie, Tits alternative for closed real analytic 4manifolds of nonpositive curvature, Topology and its applications 136 (2004), 87121.
 [X2]
 X. Xie, Groups Acting on Square Complexes, to appear in Geometriae Dedicata.
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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:
http://dx.doi.org/10.1090/S0002994704035755
PII:
S 00029947(04)035755
Keywords:
Tits boundary,
Tits metric,
$\text{CAT}(0)$,
2complex,
quasiisometry,
quasiflat
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
