The quasi-redirecting Boundary

TL;DR

This paper introduces a generalized boundary concept for metric spaces, extending Gromov boundary ideas to broader contexts, and explores its properties and relations to other boundaries, revealing new geometric insights.

Contribution

It generalizes the Gromov boundary to a larger class of metric spaces, including non-hyperbolic ones, and establishes its invariance and compatibility with existing boundary notions.

Findings

The boundary contains the sublinearly Morse boundary as a subspace.

It matches the Bowditch boundary in certain hyperbolic cases.

Reveals new QI-invariant, Morse-like quasi-geodesics in the Croke-Kleiner group.

Abstract

We generalize the notion of Gromov boundary to a larger class of metric spaces beyond Gromov hyperbolic spaces. Points in this boundary are classes of quasi-geodesic rays and the space is equipped with a topology that is naturally invariant under quasi-isometries. It turns out that this boundary is compatible with other notions of boundary in many ways; it contains the sublinearly Morse boundary as a topological subspace and it matches the Bowditch boundary of relative hyperbolic spaces when the peripheral subgroups have no intrinsic hyperbolicity. We also give a complete description of the boundary of the Croke-Kleiner group where the quasi-redirecting boundary reveals a new class of QI-invariant, Morse-like quasi-geodesics.