Sublinearly Morse Boundary I: CAT(0) Spaces

TL;DR

This paper introduces the $ abla$-Morse boundary for CAT(0) spaces, a new quasi-isometry invariant subset of the visual boundary, addressing issues with boundary definitions in non-positively curved groups.

Contribution

It defines the $ abla$-Morse boundary for CAT(0) spaces, proving its invariance under quasi-isometries and its metrizability, thus providing a well-defined boundary concept.

Findings

$ abla$-Morse boundary is QI-invariant.

$ abla$-Morse boundary is metrizable.

Poisson boundary of RAAGs identified with $ abla$-Morse boundary.

Abstract

To every Gromov hyperbolic space X one can associate a space at infinity called the Gromov boundary of X. Gromov showed that quasi-isometries of hyperbolic metric spaces induce homeomorphisms on their boundaries, thus giving rise to a well-defined notion of the boundary of a hyperbolic group. Croke and Kleiner showed that the visual boundary of non-positively curved (CAT(0)) groups is not well-defined, since quasi-isometric CAT(0) spaces can have non-homeomorphic boundaries. For any sublinear function $\kappa$, we consider a subset of the visual boundary called the $\kappa$-Morse boundary and show that it is QI-invariant and metrizable. This is to say, the $\kappa$-Morse boundary of a CAT(0) group is well-defined. In the case of Right-angled Artin groups, it is shown in the Appendix that the Poisson boundary of random walks is naturally identified with the $\sqrt{t \log t}$--boundary.