Large scale detection of half-flats in CAT(0)-spaces
Abstract
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.
Keywords
Cite
@article{arxiv.0912.1172,
title = {Large scale detection of half-flats in CAT(0)-spaces},
author = {S. Francaviglia and J. -F. Lafont},
journal= {arXiv preprint arXiv:0912.1172},
year = {2010}
}
Comments
21 pages. This article is a substantially improved version of our earlier preprint arXiv:0801.3636. It features more general results, with shorter, cleaner proofs. Applications remain the same