The Tits boundary of a 𝐶𝐴𝑇(0) 2-complex

Xie, Xiangdong

doi:10.1090/S0002-9947-04-03575-5

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

Trans. Amer. Math. Soc. 357 (2005), 1627-1661 Request permission

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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