Compressible Spaces and $mathcal{E}mathcal{Z}$-Structures

Bestvina introduced a $mathcal{Z}$-structure for a group $G$ to generalize the boundary of a CAT(0) or hyperbolic group. 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. In this paper, we show that fundamental groups of graphs of nonpositively curved Riemannian $n$-manifolds admit $mathcal{Z}$-structures and graphs of negatively curved or flat $n$-manifolds admit $mathcal{E}mathcal{Z}$-structures. This generalizes a recent result of the first two authors with Tirel, which put $mathcal{E}mathcal{Z}$-structures on Baumslag-Solitar groups and $mathcal{Z}$-structures on generalized Baumslag-Solitar groups.