Curtain Characterization of Sublinearly Morse Geodesics in CAT(0) Spaces

TL;DR

This paper demonstrates that the sublinear Morse boundary of any CAT(0) space can be continuously embedded into the Gromov boundary of a hyperbolic space, expanding understanding of boundary structures in geometric group theory.

Contribution

It introduces new curtain-based characterizations of sublinear Morse geodesics and establishes a continuous injection into hyperbolic boundaries, addressing open questions in the field.

Findings

Sublinear Morse boundary injects into hyperbolic Gromov boundary

Curtain machinery provides new characterizations of Morse geodesics

Results apply to all CAT(0) spaces, including cube complexes

Abstract

We show that the sublinear Morse boundary of every CAT(0) space continuously injects into the Gromov boundary 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. Our results answer multiple questions of Petyt-Spriano-Zalloum.