An exotic II$_1$ factor without property Gamma

TL;DR

This paper constructs a new type of II$_1$ factor that lacks property Gamma and is not elementarily equivalent to free group factors or ultraproducts of matrices, using innovative amalgamated free product techniques.

Contribution

It introduces a novel iterative amalgamated free product construction to produce explicit examples of II$_1$ factors with unique model-theoretic properties.

Findings

First explicit example of non-elementarily equivalent II$_1$ factors without property Gamma.

Constructs a II$_1$ factor not elementarily equivalent to any free group factor.

Provides a II$_1$ factor not elementarily equivalent to any ultraproduct of matrices.

Abstract

We introduce a new iterative amalgamated free product construction of II$_1$ factors, and use it to construct a separable II$_1$ factor which does not have property Gamma and is not elementarily equivalent to the free group factor $\text{L}(\mathbb F_n)$, for any $2\leq n\leq \infty$. This provides the first explicit example of two non-elementarily equivalent II$_1$ factors without property Gamma. Moreover, our construction also provides the first explicit example of a II$_1$ factor without property Gamma that is also not elementarily equivalent to any ultraproduct of matrix algebras. Our proofs use a blend of techniques from Voiculescu's free entropy theory and Popa's deformation/rigidity theory.