A class of II_1 factors with many non conjugate Cartan subalgebras

TL;DR

This paper constructs specific II_1 factors with a vast, unclassifiable diversity of Cartan subalgebras and multiple non-isomorphic group measure space decompositions, revealing complex internal symmetries.

Contribution

It introduces a class of II_1 factors with unclassifiably many Cartan subalgebras and demonstrates the existence of a single group with uncountably many non-stably orbit equivalent actions leading to isomorphic factors.

Findings

Existence of II_1 factors with unclassifiable Cartan subalgebras

Construction of a II_1 factor with uncountably many non-isomorphic decompositions

Demonstration of a group with many non-stably orbit equivalent actions

Abstract

We construct a class of II_1 factors M that admit unclassifiably many Cartan subalgebras in the sense that the equivalence relation of being conjugate by an automorphism of M is complete analytic, in particular non Borel. We also construct a II_1 factor that admits uncountably many non isomorphic group measure space decompositions, all involving the same group G. So G is a group that admits uncountably many non stably orbit equivalent actions whose crossed product II_1 factors are all isomorphic.