GEOMETRY AND DYNAMICS ON SUBLINEARLY MORSE BOUNDARIES OF CAT(0) GROUPS

Given a sublinear function κ, κ-Morse boundaries ∂_κ X of proper CAT(0) spaces are introduced by Qing, Rafi and Tiozzo (2024). It is a topological space that consists of a equivalence class of quasigeodesic rays and it is quasiisometrically invariant and metrizable. We study the sublinearly Morse boundaries with the assumption that there is a proper cocompact action of a group G on the CAT(0) space in question. We show that G acts minimally on ∂_κ G and that contracting elements of G induces a weak north-south dynamic on ∂_κ G. Also, we show that a homeomorphism f: ∂_κ G → ∂_κ G^′ comes from a quasiisometry if and only if f is successively quasimöbius and stable. Lastly, we characterize exactly when the sublinearly Morse boundary of a CAT(0) space is compact.