HNN extensions and unique group measure space decomposition of II_1 factors

TL;DR

This paper demonstrates that for a broad class of HNN extension groups, the associated group measure space II_1 factors have a unique Cartan subalgebra, leading to new examples of superrigid actions that uniquely encode the original group action.

Contribution

It establishes the uniqueness of Cartan subalgebras for certain HNN extension group measure space factors, advancing the understanding of superrigidity in operator algebras.

Findings

Unique Cartan subalgebra up to unitary conjugacy for these factors

New examples of W*-superrigid group actions

The II_1 factors fully encode the original group actions

Abstract

We prove that for a fairly large family of HNN extensions \Gamma, the group measure space II_1 factor L^\infty(X) \rtimes \Gamma given by an arbitrary free ergodic probability measure preserving action of \Gamma, has a unique group measure space Cartan subalgebra up to unitary conjugacy. We deduce from this new examples of W^*-superrigid group actions, i.e. where the II_1 factor L^\infty(X) \rtimes \Gamma entirely remembers the group action that it was constructed from.