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. For any sublinear function κ, we consider a subset of the visual boundary called the κ-Morse boundary and show that it is QI-invariant and metrizable. This 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 ( √ t log t)-boundary.

Sublinearly Morse boundaries from the viewpoint of combinatorics

“…In this section we review the definition and properties of κ-Morse geodesic rays needed for this paper. For further details, see [QRT19]. We fix a function…”

“…∞ X, is continuous. That is to say, the quasi-isometry invariant topology of Qing, Rafi and Tiozzo [QRT19] is finer than the subspace topology induced on the collection of κ-Morse geodesic rays. This is Lemma 2.12.…”

“…Several attempts have recently been made to rectify this issue, most recent of which is work of Qing, Rafi and Tiozzo where they introduce the sublinearly Morse boundary and show that it's a metrizable quasi-isometry invariant of any CAT(0) space [QRT19]. Given a sublinear function κ, a geodesic ray b is said to be κ-Morse if there exists a function m b : R + × R + → R + such that for any (q, Q)-quasi geodesic α with end points on b, if p ∈ α, we have d(p, b) ≤ m b (q, Q)κ(||p||), where ||p|| = d(p, b(0)).…”

“…(1) Denote ∂ κ X the quasi-isometry invariant κ-Morse boundary of X given in [QRT19], and ∂ κ ∞ X the subspace of the visual boundary of X consisting of κ-Morse geodesic rays with the subspace topology. We first show that the map…”

“…Cashen and Mackay introduced a refined topology for the Morse boundary (see [CM19]) and were able to show that the Morse boundary, with respect to their topology is a metrizable quasi-isometry invariant. The κ-Morse boundary was introduced by Qing, Rafi and Tiozzo in [QRT19] and sought to rectify some of the shortcomings found in the Morse boundary, for example, unlike the Morse boundary of Charney-Sultan, Cordes and Cashen-Mackay, the κ-Morse boundary serves as a topological model of the Poisson boundary for right-angled Artin groups (see [QT19] and [QRT19]). Furthermore, it was recently announced by Gekhtman, Qing and Rafi [GQR] that κ-Morse geodesics are generic in rank-1 CAT(0) spaces in various natural measures.…”