On the Conjugacy of Measurable Partitions with Respect to the Normalizer of a Full Type \(\mathrm{II}_1\) Ergodic Group

Let \(G\) be a countable ergodic group of automorphisms of a measure space \((X,\mu)\) and \(\mathcal{N}[G]\) be the normalizer of its full group \([G]\) . Problem: for a pair of measurable partitions \(\xi\) and \(\eta\) of the space \(X\) , when does there exist an element \(g\in\mathcal{N}[G]\) such that \(g\xi=\eta\) ? For a wide class of measurable partitions, we give a solution to this problem in the case where \(G\) is an approximately finite group with finite invariant measure. As a consequence, we obtain results concerning the conjugacy of the commutative subalgebras that correspond to \(\xi\) and \(\eta\) in the type \(\mathrm{II}_1\) factor constructed via the orbit partition of the group \(G\) .