Large Scale Detection of Half-flats in Cat(0) Spaces

Let M be a complete locally compact CAT(0)-space, and X an ultralimit of M . For γ ⊂M a k-dimensional flat, let γω be the k-dimensional flat in X obtained as an 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 locally compact CAT(0)-spaces, (2) constraints on the possible non-positively curved Riemannian metrics supported by certain manifolds, and (3) a correspondence between metric splittings of a complete, simply connected non-positively curved Riemannian manifolds, and metric splittings of its asymptotic cones. Furthermore, combining our results with the Ballmann, Burns-Spatzier rigidity theorem and the classic Mostow rigidity, we also obtain (4) a new proof of Gromov’s rigidity theorem for higher rank locally symmetric spaces.