Uniformly perfect Morse boundaries

In this talk, we introduce the uniformly perfect Morse boundary for proper geodesic metric spaces and give its full geometric characterization. The Morse boundary of any finitely generated non-elementary group is uniformly perfect whenever it is nonempty. We also establish a rigidity principle for homeomorphisms between such boundaries. A homeomorphism between uniformly perfect Morse boundaries is induced by a quasi-isometry of the underlying spaces if and only if it satisfies standard geometric regularity conditions. This is based on joint work with Suzhen Han.