II$_1$ factors and equivalence relations with distinct fundamental groups

TL;DR

This paper constructs a II$_1$ factor with two non-conjugate Cartan subalgebras, demonstrating a divergence between the fundamental groups of the factor and the associated equivalence relation, highlighting novel structural properties.

Contribution

It introduces a II$_1$ factor with two non-conjugate Cartan subalgebras exhibiting different fundamental groups, revealing new phenomena in operator algebra theory.

Findings

The II$_1$ factor has trivial fundamental group.

The associated equivalence relation has non-trivial fundamental group.

The second Cartan subalgebra is twisted by a 2-cocycle.

Abstract

We construct a group measure space II$_1$ factor that has two non-conjugate Cartan subalgebras. We show that the fundamental group of the II$_1$ factor is trivial, while the fundamental group of the equivalence relation associated with the second Cartan subalgebra is non-trivial. This is not absurd as the second Cartan inclusion is twisted by a 2-cocycle.