Superrigidity, profinite actions and Borel reducibility

Specialeforsvar: Vibeke Quorning

Titel: Superrigidity, profinite actions and Borel reducibility

Resume: In this thesis we study countable Borel equivalence relations, that is, orbit
equivalence relations induced by Borel actions of countable groups on standard
Borel spaces. We prove the cocycle- and orbit equivalence superrigidity theorems for free ergodic profinite actions of property (T) groups due to A. Ioana. Moreover, we will use the cocycle superrigidity theorem to obtain a necessary and sufficient condition for two free strongly ergodic translation profinite actions to be stably orbit equivalent, as well as a necessary and sufficient condition for their induced orbit equivalence relations to be
comparable with respect to Borel reducibility. Finally, we apply all these results to construct several uncountable families of pairwise stably orbit inequivalent actions whose induced orbit equivalence relations are pairwise incomparable with respect to Borel reducibility.

Vejleder: Asger Dag Törnquist
Censor: Søren Møller , SDU