On ultraproduct approximations and property (T) factors

TL;DR

This paper develops a new framework connecting deformation/rigidity theory with continuous model theory of II$_1$ factors, solving key open problems and establishing non-elementary equivalences among factors.

Contribution

It introduces a novel framework for deformation/rigidity in continuous model theory, resolving open problems and demonstrating non-elementary equivalences among II$_1$ factors.

Findings

L(SL_3(Z)) and L(F_2) are not elementarily equivalent

L(F_2) is not pseudomatricial

Constructed an infinite family of non-elementarily equivalent full factors

Abstract

We introduce a framework allowing for key aspects of deformation/rigidity theory to be used in the study of continuous model theory of II$_1$ factors. Using this framework, we solve several well-known open problems in the area. For example, we show that the group von Neumann algebras $L(SL_3(\mathbb Z))$ and $L \mathbb F_2$ are not elementarily equivalent, and we show that the group von Neumann algebra $L\mathbb F_2$ is not pseudomatricial. We also show a Bass-Serre type strong rigidity result in the setting of ultraproducts to provide an infinite family of pairwise non-elementarily equivalent full factors, each of which embeds into an ultraproduct of the hyperfinite II$_1$ factor. Building on previous work of Boutonnet, Chifan and Ioana, we also provide a continuum of pairwise non-elementarily equivalent full factors, which we can take to be group von Neumann algebras or group-measure space constructions.