Weak- Z-Structures and one-relator groups

Abstract

Motivated by the notion of boundary for hyperbolic and groups, Bestvina [2] introduced the notion of a (weak) -structure and (weak) -boundary for a group G of type (i.e., having a finite complex), with implications concerning the Novikov conjecture for G. Since then, some classes of groups have been shown to admit a weak -structure (see [15] for example), but the question whether or not every group of type admits such a structure remains open. In this paper, we show that every torsion free one-relator group admits a weak -structure, by showing that they are all properly aspherical at infinity; moreover, in the 1-ended case the corresponding weak -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.