On Uniformly Perfect Morse Boundaries

Abstract

We introduce and geometrically characterize the notion of uniformly perfect Morse boundary for proper geodesic metric spaces. As a unifying result, we prove that the Morse boundary of any finitely generated, non-elementary group is uniformly perfect whenever it is nonempty. This theorem applies to a broad class of groups, including all acylindrically hyperbolic groups, Artin groups, and hierarchically hyperbolic groups. Furthermore, we establish a rigidity theorem for homeomorphisms between such boundaries: for any two spaces with uniformly perfect Morse boundaries, a homeomorphism is induced by a quasi-isometry if and only if it satisfies any one of several natural geometric conditions. These conditions include being bi-H\"older, quasi-conformal, quasi-symmetric, or $2$-stable and quasi-M\"obius.