Rotation equivalence and cocycle superrigidity

TL;DR

This paper investigates the complexity of orbit equivalence relations induced by rational rotations on high-dimensional spheres, revealing non-treeability and non-reducibility properties, and applying superrigidity techniques.

Contribution

It introduces new results on the non-reducibility and structural properties of rotation equivalence relations in higher dimensions using superrigidity methods.

Findings

Rotation equivalence relations are not treeable in dimensions greater than 2.

In dimensions ≥5, these relations are not Borel reducible to lower dimensions.

Uncountably many pairwise incomparable equivalence relations exist under Borel reducibility.

Abstract

We analyze Euclidean spheres in higher dimensions and the corresponding orbit equivalence relations induced by the group of rational rotations from the viewpoint of descriptive set theory. It turns out that such equivalence relations are not treeable in dimension greater than $2$. Then we show that the rotation equivalence relation in dimension $n \geq 5$ is not Borel reducible to the one in any lower dimension. Our methods combine a cocycle superrigidity result from the works of Furman and Ioana with the superrigidity theorem for $S$-arithmetic groups of Margulis. We also apply our techniques to give a geometric proof of the existence of uncountably many pairwise incomparable equivalence relations up to Borel reducibility.