Sublinearly Morse boundary I: CAT(0) spaces

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.We attempt to construct an analogue of the Gromov boundary that encodes the hyperbolic directions in a metric space. To this end, for any sublinear function κ, we define a subset of the visual boundary called the κ–Morse boundary. We show that, equipped with a coarse notion of visual topology, this space is QI-invariant and metrizable. That is to say, the κ–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 (tlog⁡t)–boundary.