Weak $\mathcal Z$-structures and one-relator groups
Motivated by the notion of boundary for hyperbolic and $CAT(0)$ groups, M. Bestvina in "Local Homology Properties of Boundaries of Groups" introduced the notion of a (weak) $\mathcal Z$-structure and (weak) $\mathcal Z$-boundary for a group $G$ of type $\mathcal F$ (i.e., having a finite $K(G,1)$ complex), with implications concerning the Novikov conjecture for $G$. Since then, some classes of groups have been shown to admit a weak $\mathcal Z$-structure (see "Weak $\mathcal Z$-structures for some classes of groups" by C.R. Guilbault for example), but the question whether or not every group of type $\mathcal F$ admits such a structure remains open. In this paper, we show that every torsion free one-relator group admits a weak $\mathcal Z$-structure, by showing that they are all properly aspherical at infinity; moreover, in the $1$-ended case the corresponding weak $\mathcal Z$-boundary has the shape of either a circle or a Hawaiian earring depending on whether the group is a virtually surface group or not. Finally, we extend this result to a wider class of groups still satisfying a Freiheitssatz property.