Dottorato in MATEMATICA

Titolo della tesi: Packing conditions in metric spaces with curvature bounded above and their applications

We consider metric spaces with a synthetic notion of upper curvature bound (locally CAT(k) spaces, convex spaces and Gromov-hyoerbolic spaces). As a weak and synthetic version of lower bound on the curvature we consider a uniform packing condotion at a fixed scale. We will see how this property can be expressed in terms of upper bounds of dimension and volume of balls in the locally CAT(k) radius. Moreover it implies a quantified version of the Tits alternative on convex, Gromov-hyoerbolic metric spaces. Finally it is at the base of the equivalences between several asymptotic notions such as the covering entropy and the Minkowski dimension of the boundary