Flat sector in a proper cocompact CAT(0) space - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T03:01:09Z http://mathoverflow.net/feeds/question/86778 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/86778/flat-sector-in-a-proper-cocompact-cat0-space Flat sector in a proper cocompact CAT(0) space ckl 2012-01-27T01:15:31Z 2012-02-05T15:05:01Z <p>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 $\angle (\xi, \xi') &lt; \pi$. Using proposition 9.5 (3) of Chapter II.9 from the book <em>Metric spaces of non-positive curvature</em> 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 $\angle (\xi, \xi')$.</p> <p>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.</p> http://mathoverflow.net/questions/86778/flat-sector-in-a-proper-cocompact-cat0-space/87592#87592 Answer by Robert Bell for Flat sector in a proper cocompact CAT(0) space Robert Bell 2012-02-05T15:05:01Z 2012-02-05T15:05:01Z <p>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]. </p>