Local to global morse properties, convexity and hierarchically hyperbolic spaces

Davide Spriano (ETH, Zurich)

Abstract: In a Gromov hyperbolic space, geodesics satisfies the so-called Morse property. This means that if a geodesic and a quasi-geodesic share endpoints, then their Hausdorff distance is uniformly bounded. Remarkably, this is an equivalent characterization of hyperbolic spaces, meaning that all consequences of hyperbolicity can be ascribed to this property. Using this observation to understand hyperbolic-like behaviour in spaces which are not Gromov hyperbolic has been a very successful idea, which led to the definition of important geometric objects such as the Morse boundary and stable subgroups. Another strong consequence of hyperbolicity is the fact that local quasi-geodesics are global quasi-geodesics. This allows detecting global properties on a local scale, which has far-reaching consequences. The goal of this talk is twofold. Firstly, we will prove results that are known for hyperbolic groups in a class of spaces satisfying generalizations of the above properties. Secondly, we show that the set of such spaces is large and contains several examples of interest, i.e. CAT(0) spaces and hierarchically hyperbolic spaces.