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Abstract We extract a precise internal description of the sequential commutation equivalence relation introduced in [14] for tracial von Neumann algebras. As an application we, prove that if a tracial von Neumann algebra $$N$$ is generated by unitaries $$\{u_{i}\}_{i\in \mathbb{N}}$$ such that $$u_{i}\sim u_{j}$$ (i.e., there exists a finite set of Haar unitaries $$\{w_{i}\}_{i=1}^{n}$$ in $$N^{\mathcal{U}}$$ such that $$[u_{i}, w_{1}]= [w_{k}, w_{k+1}]=[w_{n},u_{j}]=0$$ for all $$1\leq k< n$$), then $$N$$ is singly generated. This generalizes and recovers several known single generation phenomena for II$$_{1}$$ factors in the literature with a unified proof. 

Abstract We show that for every countable group, any sequence of approximate homomorphisms with values in permutations can be realized as the restriction of a sofic approximation of an orbit equivalence relation. Moreover, this orbit equivalence relation is uniquely determined by the invariant random subgroup of the approximate homomorphisms. We record applications of this result to recover various known stability and conjugacy characterizations for almost homomorphisms of amenable groups. 

Abstract In this paper we exhibit for every non amenable group that is initially sub-amenable (sometimes also referred to as LEA), two sofic approximations that are not conjugate by any automorphism of the universal sofic group. This addresses a question of Pǎunescu and generalizes the Elek–Szabo uniqueness theorem for sofic approximations. 

We identify natural conditions for a countable group acting on a countable tree which imply that the orbit equivalence relation of the induced action on the Gromov boundary is Borel hyperfinite. Examples of this condition include acylindrical actions. We also identify a natural weakening of the aforementioned conditions that implies measure hyperfiniteness of the boundary action. We then document examples of group actions on trees whose boundary action is not hyperfinite. 

In this paper, the notion of proper proximality (introduced by Boutonnet, Ioana, and Peterson [Ann. Sci. Éc. Norm. Supér. (4) 54 (2021), 445–482]) is studied and classified in various families of groups. We show that if a group acts non-elementarily by isometries on a tree such that, for any two edges, the intersection of their edge stabilizers is finite, thenGis properly proximal. We show that the wreath productG\wr His properly proximal if and only ifHis non-amenable. We then completely classify proper proximality among graph products of non-trivial groups. Our results generalize the recent work of Duchesne, Tucker-Drob, and Wesolek classifying inner amenability for these families of groups. Our results also recover some rigidity results associated to the group von Neumann algebras by virtue of being properly proximal. A key idea in the proofs of our theorems is a technique to upgrade from relative proper proximality using computations in the double dual of the small at infinity boundary. 

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.