Sequential commutation in tracial von Neumann algebras

Recall that a unitary in a tracial von Neumann algebra is Haar if $\tau(u^n)=0$ for all $n\in \mathbb{N}$. We introduce and study a new Borel equivalence relation $\sim_N$ on the set of Haar unitaries in a diffuse tracial von Neumann algebra $N$. Two Haar unitaries $u,v$ in $\mathcal{U}(N)$ are related if there exists a finite path of sequentially commuting Haar unitaries in an ultrapower $N^\mathcal{U}$, beginning at $u$ and ending at $v$. We show that for any diffuse tracial von Neumann algebra $N$, the equivalence relation $\sim_N$ admits either 1 orbit or uncountably many orbits. We characterize property Gamma in terms of path length and number of orbits of $\sim_N$ and also show the existence of non-Gamma II$_1$ factors so that $\sim_N$ admits only 1 orbit. Examples where $\sim_N$ admits uncountably many orbits include $N$ having positive 1-bounded entropy: $h(N)>0$. As a key example, we explicitly describe $\sim_{L(\mathbb{F}_t)}$ for the free group factors. Using these ideas we introduce a numerical invariant for diffuse tracial von Neumann algebras called the commutation diameter, with applications to elementary equivalence classification. We compute lower and upper bounds for the commutation diameter in various examples. Notably we obtain non-trivial lower bounds for the family of arbitrary graph products $N$ of diffuse tracial von Neumann algebras whose underlying graph is connected and has diameter at least 4, and distinguish them up to elementary equivalence from the [CIKE23] exotic factors, despite satisfying $h(N)\leq 0$.