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

TL;DR

This paper investigates conditions under which flats in CAT(0)-spaces extend to half-flats, with implications for quasi-isometries, manifold metrics, and rigidity theorems, using ultralimits and asymptotic cone analysis.

Contribution

It introduces new conditions based on ultralimits for detecting half-flats in CAT(0)-spaces, linking geometric structures to asymptotic cones and rigidity results.

Findings

Conditions for flats to bound half-flats in CAT(0)-spaces.

Constraints on quasi-isometries between CAT(0)-spaces.

New proof of Gromov's rigidity theorem for higher rank 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.