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Transactions of the American Mathematical Society
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
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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.


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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/S0002-9947-04-03575-5
PII: S 0002-9947(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