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.