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

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 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 _ G and that contracting elements of G induces a weak north-south dynamic on _ G. We also prove, when G exists and |_ G|3, G acts minimally on G and G is a second countable topological space. The last section concerns the restriction to proper CAT (0) spaces and finite dimensional 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 G is a visibility space.