# ROOM THREE: A random matrix approach to absorption in free products

Consider a tracial free product von Neumann algebra $\mathcal{M}=\mathcal{P}*\mathcal{Q}$, where $P$ and $Q$ can be embedded into $\mathcal{R}^\omega$. Using random matrix theory, one can show that $\mathcal{P}$ contains any von Neumann subalgebra $\mathcal{N}\subset\mathcal{M}$ which intersects $\mathcal{P}$ diffusely and has 1-bounded entropy zero. Here the 1-bounded entropy of $\mathcal{N}$ is a modification of the free entropy dimension that—roughly speaking—measures how many matrix approximations $\mathcal{N}$ has. This quantity is known to be a von Neumann algebra invariant and in particular is zero for all amenable algebras. Consequently, as a corollary of the aforementioned absorption result, one obtains a novel proof of Popa’s famous theorem that the generator MASA in a free group factor is maximal amenable. In this talk I will discuss this result and some aspects of its proof. This is based on joint work with Ben Hayes, David Jekel, and Thomas Sinclair.