#### Volume 19, issue 4 (2019)

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Boundaries of Baumslag–Solitar groups

### Craig R Guilbault, Molly A Moran and Carrie J Tirel

Algebraic & Geometric Topology 19 (2019) 2077–2097
##### Abstract

A $\mathsc{Z}$–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 $\mathsc{ℰ}\mathsc{Z}$–structure. The general questions of which groups admit $\mathsc{Z}$– or $\mathsc{ℰ}\mathsc{Z}$–structures remain open. Here we show that all Baumslag–Solitar groups admit $\mathsc{ℰ}\mathsc{Z}$–structures and all generalized Baumslag–Solitar groups admit $\mathsc{Z}$–structures.

##### Keywords
$\mathcal{Z}$–structure, $\mathcal{Z}$–boundary, group boundaries, Baumslag–Solitar groups, group boundary
##### Mathematical Subject Classification 2010
Primary: 20F65, 57M07, 57M60