On the Elementary Equivalence of Group Factors

We use techniques of distortion systems, introduced by James Hanson to show that any two elementarily equivalent infinite discrete ICC groups give rise to elementarily equivalent group von Neumann algebras. This answers a question raised by Goldbring-Pi (and attributed there to Koichi Oyakawa), in the negative. In a special case, this resolves the model theoretic free factor isomorphism problem.