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We completely classify the Cartan subalgebras in all von Neumann algebras associated with graph product groups and their free ergodic measure preserving actions on probability spaces. 

We study the relationship between the dynamics of the action $$\alpha$$ of a discrete group $$G$$ on a von Neumann algebra $$M$$, and structural properties of the associated crossed product inclusion $$L(G) \subseteq M \rtimes_\alpha G$$, and its intermediate subalgebras. This continues a thread of research originating in classical structural results for ergodic actions of discrete, abelian groups on probability spaces. A key tool in the setting of a noncommutative dynamical system is the set of quasinormalizers for an inclusion of von Neumann algebras. We show that the von Neumann algebra generated by the quasinormalizers captures analytical properties of the inclusion $$L(G) \subseteq M \rtimes_\alpha G$$ such as the Haagerup Approximation Property, and is essential to capturing ``almost periodic" behavior in the underlying dynamical system. Our von Neumann algebraic point of view yields a new description of the Furstenberg-Zimmer distal tower for an ergodic action on a probability space, and we establish new versions of the Furstenberg-Zimmer structure theorem for general, tracial $W^*$-dynamical systems. We present a number of examples contrasting the noncommutative and classical settings which also build on previous work concerning singular inclusions of finite von Neumann algebras. 

We introduce a new iterative amalgamated free product construction of II factors, and use it to construct a separable II factor which does not have property Gamma and is not elementarily equivalent to the free group factor $$L(F_n)$$, for any $$n\geq 2$$. This provides the first explicit example of two non-elementarily equivalent $$II_1$$ factors without property Gamma. Moreover, our construction also provides the first explicit example of a $$II_1$$ factor without property Gamma that is also not elementarily equivalent to any ultraproduct of matrix algebras. Our proofs use a blend of techniques from Voiculescu’s free entropy theory and Popa’s deformation/rigidity theory. 

We introduce a new class of groups called {\it wreath-like products}. These groups are close relatives of the classical wreath products and arise naturally in the context of group theoretic Dehn filling. Unlike ordinary wreath products, many wreath-like products have Kazhdan's property (T). In this paper, we prove that any group $$G$$ in a natural family of wreath-like products with property (T) is W$^*$-superrigid: the group von Neumann algebra $$\text{L}(G)$$ remembers the isomorphism class of $$G$$. This allows us to provide the first examples (in fact, $$2^{\aleph_0}$$ pairwise non-isomorphic examples) of W$^*$-superrigid groups with property (T).