Weak $\mathcal Z$-structures and one-relator groups

TL;DR

This paper proves that all torsion-free one-relator groups admit a weak $ ext{Z}$-structure, revealing their boundary shapes and extending results to groups with the Freiheitssatz property, advancing understanding of group boundaries.

Contribution

It demonstrates that torsion-free one-relator groups are properly aspherical at infinity and characterizes their weak $ ext{Z}$-boundaries, extending the class of groups known to admit such structures.

Findings

Torsion-free one-relator groups admit weak $ ext{Z}$-structures.

Weak $ ext{Z}$-boundaries are circles or Hawaiian earrings.

Results extend to groups with the Freiheitssatz property.

Abstract

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.