The Curtain Model of CAT(0) Spaces and its Relationship to the Sublinearly Morse Boundary
Abstract
We show that the sublinear Morse boundary of every CAT(0) space continuously injects into the Gromovboundary of a hyperbolic space, which was not previously known even for all CAT(0) cube complexes. Our work utilizes the curtain machinery introduced by Petyt-Spriano-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. The hyperbolic space mentioned is the curtain model, and its role for a CAT(0) space has shown a striking comparison to the curve graph for a mapping class group of a finite type surface. We show that the curtain model is not a quasi-isometry invariant for all CAT(0) spaces and that quasi-flats are of bounded diameter in the curtain model. Our results answer multiple questions of Petyt-Spriano-Zalloum.