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Let $X$ be a complete CAT(0) space with a proper and cocompact group action by isometries, and suppose there are $\xi, \xi' \in \partial X$ with $\angle (\xi, \xi') < \pi$. Using proposition 9.5 (3) of Chapter II.9 from the book Metric spaces of non-positive curvature by Bridson and Haefliger, there exist two points $\\eta, \eta' \in \partial X$ and a point $y\in X$ such that $\angle_y (\eta,\eta') =\angle(\xi,\xi')$. Then by corollary 9.9 of the same chapter, the convex hull of two geodesic rays $c, c'$ issuing from $y$ with $c(\infty) = \eta$, $c'(\infty) = \eta'$ is isometric to a sector in $\mathbb{E}^2$ bounded by two rays meeting at the angle $\angle (\xi, \xi')$.

The above result gives geodesic rays with possibly different endpoints. So I would like to know whether we can have a stronger result that there are geodesic rays $c_1, c'_1$ issuing from some point $y_1\in X$ with endpoints $\xi,\xi'$ such that the convex hull of these rays is isometric to a flat sector. I think this should be true in general, but maybe there is a counter-example that I do not know.

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I think one can prove that there is a isometric embedding of a round circle into $\partial X$ containing $\xi, \xi'$ with respect to the Tits metric (there is a unique isometric arc connecting $\xi, \xi'$ of length $\angle(\xi, \xi')$, which will be an arc of this circle. So I think one could answer your question if you could prove that there is a geodesic connecting points of Tits distance $=\pi$. –  Ian Agol Jan 28 '12 at 7:07
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up vote 2 down vote accepted

Xie proved that this is true if $X$ is a 2 dimensional complex. I recommend reading his article: "The Tits Boundary of a CAT(0) 2-Complex", Trans. AMS., 357, no. 4, 1627-1661. I think that the answer to the more general question above is unknown. The techniques used in Xie's paper are further developed in an article by Bestvina, Kleiner, and Sageev: "Quasi-flats in CAT(0) complexes", arXiv:0804.2619v1 [math.GR].

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