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

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

2009-12-07

arxiv.org/abs/0912.1172v1

Indiana Univ. Math. J. 59 (2010), pgs. 395-415

Mathematics

Metric Geometry

21 pages. This article is a substantially improved version of our earlier preprint arXiv:0801.3636. It features more general r

Scientific paper

For a k-flat F inside a locally compact CAT(0)-space X, we identify various conditions that ensure that F bounds a (k+1)-dimensional half flat in X. Our conditions are formulated in terms of the ultralimit of X. As applications, we obtain (1) constraints on the behavior of quasi-isometries between tocally 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 manifold and the metric splittings of its asymptotic cones. Furthermore, combining our results with the Ballmann, Burns-Spatzier rigidity theorem and the classical Mostow rigidity theorem, we also obtain (4) a new proof of Gromov's rigidity theorem for higher rank locally symmetric spaces.

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