Title: Large scale detection of half-flats in CAT(0)-spaces

Authors: Stefano Francaviglia and Jean-Francois Lafont

Issue: Volume 59 (2010), Issue 2, 395-416

Abstract: Let $M$ be a complete locally compact $\mbox{CAT}(0)$-space, and $X$ an asymptotic cone of $M$. For $\gamma \subset M$ a $k$-dimensional flat, let $\gamma_{\omega}$ be the $k$-dimensional flat in $X$ obtained as the ultralimit of $\gamma$. In this paper, we identify various conditions on $\gamma_{\omega}$ that are sufficient to ensure that $\gamma$ bounds a $(k+1)$-dimensional half-flat. As applications we obtain: (1) constraints on the behavior of quasi-isometries between locally compact $\mbox{CAT}(0)$-spaces; (2) constraints on the possible non-positively curved Riemannian metrics supported by certain manifolds; (3) a correspondence between metric splittings of a complete, simply connected non-positively curved Riemannian manifolds, and metric splittings of its asymptotic cones; and (4) an elementary derivation of Gromov's rigidity theorem from the combination of the Ballmann, Burns-Spatzier rank rigidity theorem and the classic Mostow rigidity theorem.