Elaborating on Joel David Hamkin's comment---Simon Thomas has provided applications of various super-ridigity theorems (from the ergodic theory of group actions) to the theory of the Borel complexity of countable equivalence relations, for example he shows that the universal countable equivalence relation is not essentially free

Thomas, Simon, Popa superrigidity and countable Borel equivalence relations, Ann. Pure Appl. Logic 158, No. 3, 175-189 (2009). ZBL1162.03029.

Inspired by Joel David Hamkin's comment---Simon Thomas has provided applications of various super-ridigity theorems (from the ergodic theory of group actions) to the theory of the Borel complexity of countable equivalence relations, for example he shows that the universal countable equivalence relation is not essentially free

Thomas, Simon, Popa superrigidity and countable Borel equivalence relations, Ann. Pure Appl. Logic 158, No. 3, 175-189 (2009). ZBL1162.03029.