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

TL;DR

This paper demonstrates that certain fundamental groups of manifolds with nonpositive or negative curvature admit $ ext{Z}$-structures or $ ext{EZ}$-structures, extending previous results to broader classes of groups.

Contribution

It establishes the existence of $ ext{Z}$- and $ ext{EZ}$-structures for fundamental groups of specific curved manifolds, generalizing earlier work on Baumslag-Solitar groups.

Findings

Fundamental groups of nonpositively curved manifolds admit $ ext{Z}$-structures.

Fundamental groups of negatively curved or flat manifolds admit $ ext{EZ}$-structures.

Extends previous results on Baumslag-Solitar groups to broader classes.

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. 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.