On Uniformly Perfect Morse Boundaries

TL;DR

This paper introduces the concept of uniformly perfect Morse boundaries for proper geodesic metric spaces and proves their properties and rigidity under homeomorphisms, with applications to various classes of groups.

Contribution

It defines uniformly perfect Morse boundaries, proves their uniform perfectness for a wide class of groups, and establishes a rigidity theorem linking boundary homeomorphisms to quasi-isometries.

Findings

Morse boundary of finitely generated groups is uniformly perfect when nonempty.

Homeomorphisms between such boundaries are induced by quasi-isometries under geometric conditions.

The results apply to acylindrically hyperbolic, Artin, and hierarchically hyperbolic groups.

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.