Compressible spaces and $\mathcal{E}\mathcal{Z}$-structures
Volume 256 / 2022
Abstract
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. A recent result of the first two authors with Tirel put $\mathcal {E}\mathcal {Z}$-structures on Baumslag–Solitar groups and $\mathcal {Z}$-structures on generalized Baumslag–Solitar groups. We generalize this to higher dimensions by showing that fundamental groups of graphs of closed nonpositively curved Riemannian $n$-manifolds (each vertex and edge manifold is of dimension $n$) admit $\mathcal {Z}$-structures, and graphs of negatively curved or flat Riemannian $n$-manifolds admit $\mathcal {E}\mathcal {Z}$-structures.
