# Algebra Seminar talk

2014-10-24

*Fabio*

**Tonti**Abstract:

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.