Concentration Week in Set Theory and Functional Analysis, College Station, August 2010

Abstract: Borel reducibility is a tool from descriptive set theory which allows one to compare the complexity of classification problems from algebra, analysis and logic. Since many classification problems are captured by equivalence relations induced by a natural group action, this area sometimes resembles orbit equivalence theory.

In this talk, we will consider the relation induced by the action of $SL_n(\mathbb Z)$ on $SL_n(\mathbb Z_p)$ (over the $p$-adics). Motivated by a connection with torsion-free abelian groups, Hjorth–Thomas essentially showed that as $p$ varies, the corresponding relations are incomparable with respect to Borel reducibility. The proof requires powerful tools: either Zimmer’s superrigidity theorem for lattices in Lie groups, or Ioana’s recent superrigidity theorem for profinite actions. I’ll indicate how this works, and time permitting, give some newer incomparability results.