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

TL;DR

This paper investigates the properties of sublinearly Morse boundaries in CAT(0) spaces under group actions, revealing their topological dynamics, invariance under quasi-isometries, and conditions for compactness.

Contribution

It establishes minimal group actions, dynamic behaviors, and characterizes when sublinearly Morse boundaries are compact, advancing understanding of geometric group theory.

Findings

Group $G$ acts minimally on $ ext{p}_ ext{sub} G$

Contracting elements induce weak north-south dynamics

Characterization of compactness of sublinearly Morse boundaries

Abstract

Given a sublinear function $\kappa$, $\kappa$-Morse boundaries $\pka X$ of proper \CAT spaces are introduced by Qing, Rafi and Tiozzo. It is a topological space that consists of a large set of quasi-geodesic rays and it is quasi-isometrically invariant and metrizable. In this paper, we study the sublinearly Morse boundaries with the assumption that there is a proper cocompact action of a group $G$ on the \CAT space in question. We show that $G$ acts minimally on $\pka G$ and that contracting elements of $G$ induces a weak north-south dynamic on $\pka G$. Furthermore, we show that a homeomorphism $f \from \pka G \to \pka G'$ comes from a quasi-isometry if and only if $f$ is successively quasi-m{\"o}bius and stable. Lastly, we characterize exactly when the sublinearly Morse boundary of a \CAT space is compact.