Games and elementary equivalence of $\rm II_1$ factors

TL;DR

This paper introduces a geometric criterion using Ehrenfeucht-Fra"issé games to determine elementary equivalence of II$_1$ factors, linking it to the elementary equivalence of their unitary groups as $ ext{Z}_4$-metric spaces.

Contribution

It provides a novel geometric approach to characterize elementary equivalence of II$_1$ factors via their unitary groups.

Findings

Elementary equivalence of II$_1$ factors is characterized by their unitary groups.

A local geometric criterion for elementary equivalence is established.

Elementary equivalence of unitary groups as $ ext{Z}_4$-metric spaces implies that of the factors.

Abstract

We use Ehrenfeucht-Fra\"iss\'e games to give a local geometric criterion for elementary equivalence of II$_1$ factors. We obtain as a corollary that two II$_1$ factors are elementarily equivalent if and only their unitary groups are elementarily equivalent as $\mathbb Z_4$-metric spaces.