Curtain characterization of sublinearly Morse geodesics in $\mathrm{CAT}(0)$ spaces

Abstract

We show that the sublinear Morse boundary of every $\mathrm{CAT}(0)$ space continuously injects into the Gromov boundary of a hyperbolic space, which was not previously known even for all $\mathrm{CAT}(0)$ cube complexes. Our work utilizes the curtain machinery introduced by Petyt, Spriano, and Zalloum. Curtains are more general combinatorial analogues of hyperplanes in cube complexes, and we develop multiple curtain characterizations of the sublinear Morse property along the way. Our results answer multiple questions of Petyt, Spriano, and Zalloum.