Abstract

In this talk I will first survey several classes of metric spaces that are non-positively curved in a (very) weak global sense and explain how they are interrelated. A focus will be on coarsely injective spaces, which have recently proven to play an unexpectedly universal role in generalizations of Gromov hyperbolicity. In a second part I will discuss some joint work with Tommaso Goldhirsch which complements results of Stefan Wenger and of Bruce Kleiner and myself on hyperbolicity phenomena in spaces of higher asymptotic rank.