Borel complexity of the isomorphism relation of Archimedean orders in finitely generated groups
Antoine Poulin · Proceedings of the American Mathematical Society · 2025
In 2020, Calderoni, Marker, Motto Ros and Shani asked what the Borel complexity of the isomorphism relation of Archimedean orders on Q n \mathbb {Q}^n is. We answer this question by proving that the isomorphism relation of Archimedean orders on Z n \mathbb {Z}^n is not hyperfinite when n ≥ 3 n \geq 3 and not treeable when n ≥ 4 n \geq 4 . As a corollary, we get that the isomorphism relation of Archimedean orders on Q n \mathbb {Q}^n is not hyperfinite when n ≥ 3 n \geq 3 and not treeable when n ≥ 4 n \geq 4 .