Weak Z-structures and one-relator groups

Abstract

Motivated by the notion of boundary for hyperbolic and C A T ( 0 ) groups, Bestvina [2] 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 [15] 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.