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

Yulan Qing and Abdul Zalloum

Vol. 333 (2024), No. 1, 155–179

DOI: 10.2140/pjm.2024.333.155

Abstract

Given a sublinear function $\kappa$, $\kappa$-Morse boundaries ${\partial }_{\kappa }X$ of proper $\mathrm{CAT}<!--FUNCTION\; APPLICATION-->(0)$ 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 $\mathrm{CAT}<!--FUNCTION\; APPLICATION-->(0)$ space in question. We show that $G$ acts minimally on  ${\partial }_{\kappa }G$ and that contracting elements of $G$ induces a weak north-south dynamic on ${\partial }_{\kappa }G$. Also, we show that a homeomorphism $f:{\partial }_{\kappa }G\to {\partial }_{\kappa }{G}^{\prime }$ 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 $\mathrm{CAT}<!--FUNCTION\; APPLICATION-->(0)$ space is compact.

Keywords

minimality, sublinearly Morse boundary, compact, quasimöbius

Mathematical Subject Classification

Primary: 20F65

Milestones

Received: 7 January 2023

Revised: 24 July 2024

Accepted: 30 August 2024

Published: 19 December 2024