Algebra Seminar talk
We give an overview of some structural results concerning Borel reducibility on countable Borel equivalence relations. We first show how to produce continuum many incomparable countable Borel equivalence relations, by using rigid group actions which are provided by Sorin Popa's superrigidity theorem. We then turn to a discussion of the classification problem for torsion free abelian groups of finite rank and present Simon Thomas's solution of it. Finally, we show that this classification problem is not countable universal.