Geometry and dynamics on sublinearly Morse boundaries of CAT(0) groups (Q6655548)

Let \(G\) be an hyperbolic group. The collection of all infinite geodesic rays in the associated Cayley graph of \(G\), equipped this set with cone topology, is the space boundary \(\partial G\). \textit{Y. Qing} and \textit{K. Rafi}, in [Adv. Math. 404, Part B, Article ID 108442, 51 p. (2022; Zbl 1512.20143)], introduced the sublinearly Morse boundary \(\partial_{\kappa} X\) of a CAT(0) metric space \(X\) and show that \(\partial_{\kappa} X\) is quasiisometry invariant and metrizable.\N\NIn the paper under review the authors 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. They show that \(G\) acts minimally on \(\partial_{\kappa} G\) and that contracting elements of \(G\) induces a weak north-south dynamic on \(\partial_{\kappa}G\). Moreover they show that a homeomorphism \(f: \partial_{\kappa} G \rightarrow \partial_{\kappa} G'\) comes from a quasiisometry if and only if \(f\) is successively quasi-Möbius and stable. Finally they characterize exactly when the sublinearly Morse boundary of a CAT(0) space is compact.