On Uniformly Perfect Morse Boundaries

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-Hölder, quasi-conformal, quasi-symmetric, or $2$-stable and quasi-Möbius.

💡 Research Summary

The paper introduces a notion of “uniformly perfect” Morse boundary for proper geodesic metric spaces and provides a comprehensive geometric characterization of this property. The authors first adapt the classical definition of uniform perfectness—originally formulated for compact metric spaces—to the intrinsically non‑metrizable setting of Morse boundaries by using Hausdorff distances between Morse rays (Definition 3.4). They then prove that three seemingly different geometric conditions are equivalent: (1) the Morse boundary ∂*X is uniformly perfect and X is uniformly Morse‑based; (2) X is Morse‑geodesically rich, meaning that Morse geodesics are abundant enough to connect arbitrary points with uniformly controlled Morse gauges; (3) X is center‑exhaustive, i.e., every finite‑radius ball can be enlarged to eventually capture the “center” of the boundary. This equivalence is the content of Theorem 1.1 and forms the backbone of the paper.

For co‑compact proper spaces, Theorem 1.2 shows that uniform perfectness automatically implies Morse‑boundary rigidity: the natural homomorphism from the quasi‑isometry group QI(X) to Homeo(∂*X) is injective. This rigidity mirrors the classical situation for Gromov boundaries of hyperbolic spaces but now holds in the broader Morse setting.

A key algebraic consequence follows from a known result (Corollary 5.8 in