Boundaries for geodesic spaces

TL;DR

This paper introduces a new quasi-geometric boundary for proper geodesic spaces, which generalizes the Gromov boundary and is invariant under quasi-isometries, providing a unified boundary concept.

Contribution

It defines the quasi-geometric boundary for proper geodesic spaces and establishes its key properties, including invariance under quasi-isometries and its relation to Gromov hyperbolic spaces.

Findings

The boundary is compact and metric.

It coincides with the Gromov boundary for hyperbolic spaces.

It is trivial for Croke-Kleiner spaces.

Abstract

For every proper geodesic space $X$ we introduce its quasi-geometric boundary $\partial_{QG}X$ with the following properties: 1. Every geodesic ray $g$ in $X$ converges to a point of the boundary $\partial_{QG}X$ and for every point $p$ in $\partial_{QG}X$ there is a geodesic ray in $X$ converging to $p$, 2. The boundary $\partial_{QG}X$ is compact metric, 3. The boundary $\partial_{QG}X$ is an invariant under quasi-isometric equivalences, 4. A quasi-isometric embedding induces a continuous map of quasi-geodesic boundaries, 5. If $X$ is Gromov hyperbolic, then $\partial_{QG}X$ is the Gromov boundary of $X$. 6. If $X$ is a Croke-Kleiner space, then $\partial_{QG}X$ is a point.