Many recent results in the theory of measure preserving group actions and equivalence relations that do not {\it a priori} involve von Neumann algebras were obtained using von Neumann algebraic techniques. This talk will focus on further examples of this phenomenon in two directions. First, we study unique factorization for products of measure preserving equivalence relations $\mathcal{R}_1 \times \mathcal{R}_2 \times \cdots \times \mathcal{R}_k$. Second, we will discuss a joint work with Lewis Bowen and Adrian Ioana, which concerns properties of general non-amenable groups and equivalence relations that can deduced from properties of a prototypical non-amenable group $\mathbb{F}_2$, the free group on two generators.