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

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.

v2: Some minor adjustments were made. Accepted for publication in International Journal of Mathematics

References in corpus (1)

• Explicit examples of equivalence relations and factors with prescribed fundamental group and outer automorphism group