Quasi-redirecting boundaries of non-positively curved groups

Abstract

The quasi-redirecting (QR) boundary is a close generalization of the Gromov boundary to all finitely generated groups. In this paper, we establish that the QR boundary exists as a topological space for several well-studied classes of groups. These include fundamental groups of irreducible non-geometric 3-manifolds, groups that are hyperbolic relative to subgroups with well-defined QR boundaries, right-angled Artin groups whose defining graphs are trees, and right-angled Coxeter groups whose defining flag complexes are planar. This result significantly broadens the known existence of QR boundaries. Additionally, we give a complete characterization of the QR boundaries of Croke-Kleiner admissible groups that act geometrically on CAT(0) spaces. We show that these boundaries are non-Hausdorff and can be understood as one-point compactifications of the Morse-like directions. Finally, we prove that if G is hyperbolic relative to subgroups with well-defined QR boundaries, then the QR boundary of G maps surjectively onto its Bowditch boundary.