Boundaries for geodesic 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.

15 pages