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

Let M be a complete locally compact CAT(0)-space, and X an asymptotic cone of M . For γ ⊂ M a k-dimensional flat, let γω be the k-dimensional flat in X obtained as the ultralimit of γ . In this paper, we identify various conditions on γω that are sufficient to ensure that γ bounds a (k+ 1)-dimensional half-flat. As applications we obtain: (1) constraints on the behavior of quasi-isometries between lo- cally compact CAT(0)-spaces; (2) constraints on the possible non-positively curved Riemann- ian metrics supported by certain manifolds; (3) a correspondence between metric splittings of a complete, simply connected non-positively curved Riemannian mani- folds, 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 rigid- ity theorem and the classic Mostow rigidity theorem.