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

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.