Characterizations of higher rank hyperbolicity

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 inequality for 1-cycles, 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.