Characterizations of higher rank hyperbolicity

TL;DR

This paper explores the properties of higher rank hyperbolicity, establishing six equivalent conditions in the context of generalized non-positive curvature and higher asymptotic rank, extending classical hyperbolicity concepts.

Contribution

It provides a comprehensive list of six equivalent properties for higher rank hyperbolicity, with a self-contained proof and improvements over existing results.

Findings

Six equivalent properties for higher rank hyperbolicity established

Simplified and improved proofs of known results provided

Extension of classical hyperbolicity concepts to higher rank settings

Abstract

The concept of Gromov hyperbolicity manifests itself in many different ways. With only mild assumptions on the underlying metric space, the spectrum of equivalent properties includes various thin triangle conditions, the stability of quasi-geodesics (the Morse lemma), a linear isoperimetric filling inequality for closed curves, and a sub-quadratic isoperimetric inequality. We present a similar list of six equivalent properties in the context of generalized non-positive curvature and higher asymptotic rank. This complements results of Wenger and of Kleiner and the second author. We give a largely self-contained proof, providing some improvements and simplifications for the known part.