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Abstract: The theory of countable Borel equivalence relations analyzes the actions of countable groups on Polish spaces. The main question studied is how much information is encoded by the corresponding orbit space. The amount of encoded information reflects the extent to which the action is rigid.

In this talk, we will discuss rigidity results for the action of the group of rational rotations on spheres in higher dimension. These are connected to superrigidity results of Margulis and to Zimmer's program about the actions of discrete subgroups of Lie groups on manifolds. Moreover, our methods provide new examples of countable Borel equivalence relations, and give a new proof of a fundamental theorem of Adams and Kechris about Borel complexity.