Boundaries of Baumslag-Solitar Groups

A $\mathcal{Z}$-structure on a group $G$ was introduced by Bestvina in order to extend the notion of a group boundary beyond the realm of CAT(0) and hyperbolic groups. A refinement of this notion, introduced by Farrell and Lafont, includes a $G$-equivariance requirement, and is known as an $\mathcal{E}\mathcal{Z}$-structure. The general questions of which groups admit $\mathcal{Z}$- or $\mathcal{E}\mathcal{Z}$-structures remain open. In this paper we add to the current knowledge by showing that all Baumslag-Solitar groups admit $\mathcal{E}\mathcal{Z}$-structures and all generalized Baumslag-Solitar groups admit $\mathcal{Z}$-structures.