Asymptotic rank of spaces with bicombings

TL;DR

This paper extends the understanding of when quasiflats in metric spaces imply the existence of flats, generalizing results to spaces with bicombings, a weak form of non-positive curvature, using metric differentiation and barycenter techniques.

Contribution

It generalizes Kleiner's theorem to spaces with bicombings, broadening the class of spaces where quasiflat implies flat, without requiring unique geodesics.

Findings

Generalization of flatness results to bicombing spaces

Use of barycenter construction for metric differentiation

Establishment of conditions linking quasiflats and flats

Abstract

The question, under what geometric assumptions on a space X an n-quasiflat in X implies the existence of an n-flat therein, has been investigated for a long time. It was settled in the affirmative for Busemann spaces by Kleiner, and for manifolds of non-positive curvature it dates back to Anderson and Schroeder. We generalize the theorem of Kleiner to spaces with bicombings. This structure is a weak notion of non-positive curvature, not requiring the space to be uniquely geodesic. Beside a metric differentiation argument, we employ an elegant barycenter construction due to Es-Sahib and Heinich by means of which we define a Riemannian integral serving us in a sort of convolution operation.