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

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

Published electronically:
October 28, 2004

MathSciNet review:
2115379

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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.

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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:
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