Borel Complexity of the Isomorphism Relation of Archimedean Orders in Finitely Generated Groups

In 2020, Calderoni, Marker, Motto Ros and Shani asked what the Borel complexity of the isomorphism relation of Archimedean orders on ${\mathbb{Q}}^n$ is. We answer this question by proving that the isomorphism relation of Archimedean orders on ${\mathbb{Z}}^n$ is not hyperfinite when $n\ge 3$ and not treeable when $n\ge 4$. As a corollary, we get that the isomorphism relation of Archimedean orders on ${\mathbb{Q}}^n$ is not hyperfinite when $n\ge 3$ and not treeable when $n\ge 4$.