A tree-free group that is not orderable

TL;DR

This paper constructs a specific group acting freely on a $bZ^2$-tree that contains non-trivial torsion elements, demonstrating it is not orderable, thus answering a question about groups acting on $bZ$-trees.

Contribution

It provides the first example of a group acting freely on a $bZ^2$-tree that is not orderable due to the presence of torsion elements.

Findings

The group acts freely on a $bZ^2$-tree.

The group contains non-trivial generalized torsion elements.

The group is not orderable.

Abstract

I. M. Chiswell has asked whether every group that admits a free isometric action (without inversions) on a $\Lambda$-tree is orderable. We give an example of a multiple HNN extension $\Gamma$ which acts freely on a $\mathbb{Z}^2$-tree but which has non-trivial generalised torsion elements. The existence of such elements implies that $\Gamma$ is not orderable.