Uniform Perfectness, Geodesic Richness, and Rigidity for Sublinearly Morse Boundaries

Han and Liu gave a geometric characterization of uniform perfectness for the Morse boundary of a proper geodesic metric space: the Morse boundary is uniformly perfect if and only if the space is Morse geodesically rich, equivalently center–exhaustive. In this paper we prove the analogous statement for the sublinearly Morse boundary ${\mathrm{\partial }}_{\kappa }X$. Here $\kappa$ is a fixed concave increasing sublinear function and ${\mathrm{\partial }}_{\kappa }X$ is the boundary introduced by Qing–Rafi for CAT(0) spaces and extended by Qing–Rafi–Tiozzo to proper geodesic spaces. Assuming that ${\mathrm{\partial }}_{\kappa }X$ has at least three points, we show that uniform perfectness of ${\mathrm{\partial }}_{\kappa }X$ (for any $\kappa$–visual metric based at a fixed basepoint) is equivalent to $\kappa$–Morse geodesic richness and to $\kappa$–center–exhaustiveness. The geometric input is a sublinear thin–triangle statement for $\kappa$–Morse geodesics, together with the renormalization map ${\rho }_{\kappa }(t)={\int }_0^t\frac{ds}{\kappa (s)}$, which converts $\kappa$–scale errors at radius $R$ into bounded errors in the ${\rho }_{\kappa }$–scale. As applications we obtain quantitative lower bounds on the lower Assouad dimension (and, under doubling hypotheses, on the Hausdorff dimension) of $\kappa$–visual metrics on ${\mathrm{\partial }}_{\kappa }X$ in terms of the uniform perfectness constant. Finally, for $\kappa$–center–exhaustive spaces $X$ and $Y$ satisfying a mild additional growth condition on $\kappa$, we prove a rigidity statement in the sublinear category: every quasi-symmetric homeomorphism ${\mathrm{\partial }}_{\kappa }X\to {\mathrm{\partial }}_{\kappa }Y$ is induced by a sublinear bilipschitz equivalence $X\to Y$.