On the Elementary Equivalence of Group Factors

TL;DR

This paper demonstrates that for infinite discrete ICC groups, elementary equivalence implies their von Neumann algebras are also elementarily equivalent, providing a negative answer to a previously posed question and resolving a specific free factor isomorphism problem.

Contribution

It introduces the use of distortion systems to connect elementary equivalence of groups with that of their von Neumann algebras, addressing open questions in the field.

Findings

Elementary equivalence of groups implies elementary equivalence of their von Neumann algebras.

Provides a negative answer to a question by Goldbring-Pi about the relationship between group and algebra elementary equivalence.

Resolves the model theoretic free factor isomorphism problem in a special case.

Abstract

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.