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Compressible spaces and $\mathcal{E}\mathcal{Z}$-structures

Volume 256 / 2022

Craig Guilbault, Molly Moran, Kevin Schreve Fundamenta Mathematicae 256 (2022), 47-75 MSC: Primary 20F36, 20F55, 20F65, 57S30, 57Q35; Secondary 20J06, 32S22. DOI: 10.4064/fm972-7-2021 Published online: 18 November 2021


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.


  • Craig GuilbaultDepartment of Mathematical Sciences
    University of Wisconsin-Milwaukee
    Milwaukee, WI 53201, USA
  • Molly MoranDepartment of Mathematics
    The Colorado College
    Colorado Springs, CO 80903, USA
  • Kevin SchreveDepartment of Mathematics
    Louisiana State University
    Baton Rouge, LA 70803-4918, USA

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