  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.
