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) Z-structure and (weak) Z-boundary for
a group G of type 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 Z-structure (see
"Weak Z-structures for some classes of groups" by C.R. Guilbault for
example), but the question whether or not every group of type F
admits such a structure remains open. In this paper, we show that every torsion
free one-relator group admits a weak Z-structure, by showing that
they are all properly aspherical at infinity; moreover, in the 1-ended case
the corresponding weak 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