A non-uniformly inner amenable group

Abstract

We provide an example of two elementarily equivalent countable ICC groups $G$ and $H$ such that $G$ is amenable and $H$ is not inner amenable. As a result, we provide the first example of elementarily equivalent groups whose group von Neumann algebras are not elementarily equivalent, answering a question asked by many researchers.

arXiv comment8 pages; third version; Question 4.6 in second version answered affirmatively; comments welcome!