Boundaries of Baumslag–Solitar groups

Abstract

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. Here we show that all Baumslag–Solitar groups admit $\mathcal{ℰ}\mathcal{Z}$–structures and all generalized Baumslag–Solitar groups admit $\mathcal{Z}$–structures.