Topological and Dynamic Properties of the Sublinearly Morse Boundary and the Quasi-Redirecting Boundary

TL;DR

This paper explores the properties of sublinearly Morse and quasi-redirecting boundaries in proper geodesic spaces, focusing on group actions, minimality, dynamics, and implications for CAT(0) spaces and cube complexes.

Contribution

It extends the understanding of these boundaries by analyzing group actions, minimality, and dynamics, and links boundary properties to the existence of Morse elements in CAT(0) spaces.

Findings

G acts minimally on ppa G boundary.

Contracting elements induce weak north-south dynamics.

Existence of QR boundary implies Morse elements in CAT(0) cube complexes.

Abstract

Sublinearly Morse boundaries of proper geodesic spaces are introduced by Qing, Rafi and Tiozzo. Expanding on this work, Qing and Rafi recently developed the quasi-redirecting boundary, denoted $\partial G$, to include all directions of metric spaces at infinity. Both boundaries are topological spaces that consist of equivalence classes of quasi-geodesic rays and are quasi-isometrically invariant. In this paper, we study these boundaries when the space is equipped with a geometric group action. In particular, we show that $G$ acts minimally on $\partial_\kappa G$ and that contracting elements of G induces a weak north-south dynamic on $\partial_\kappa G$. We also prove, when $\partial G$ exists and $|\partial_\kappa G|\geq3$, $G$ acts minimally on $\partial G$ and $\partial G$ is a second countable topological space. The last section concerns the restriction to proper CAT(0) spaces and finite dimensional \CAT cube complexes. We show that when $G$ acts geometrically on a finite dimensional CAT(0) cube complex (whose QR boundary is assumed to exist), then a nontrivial QR boundary implies the existence of a Morse element in $G$. Lastly, we show that if $X$ is a proper cocompact CAT(0) space, then $\partial G$ is a visibility space.