Rotation equivalence and cocycle superrigidity

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