A class of II_1 factors with exactly two group measure space decompositions

TL;DR

This paper constructs specific II_1 factors with a precise number of group measure space decompositions, demonstrating exact counts of such decompositions up to conjugacy.

Contribution

It introduces the first examples of II_1 factors with exactly two or n group measure space decompositions, advancing understanding of their structural uniqueness.

Findings

Constructed II_1 factors with exactly two decompositions

Extended construction to factors with exactly n decompositions

Provided new insights into the uniqueness of group measure space decompositions

Abstract

We construct the first II_1 factors having exactly two group measure space decompositions up to unitary conjugacy. Also, for every positive integer $n$, we construct a II_1 factor $M$ that has exactly $n$ group measure space decompositions up to conjugacy by an automorphism.