Simplices of maximally amenable extensions in II$_1$ factors

Abstract: For every $n\in \mathbb{N}$ we obtain a separable II$_1$ factor $M$ and a maximally abelian subalgebra $A\subset M$ such that the space of maximally amenable extensions of $A$ in $M$ is affinely identified with the $n$ dimensional $\mathbb{R}$-simplex. This moreover yields first examples of masas in II$_1$ factors $A\subset M$ admitting exactly $n$ maximally amenable factorial extensions. Our examples of such $M$ are group von Neumann algebras of free products of lamplighter groups amalgamated over the acting group. A conceptual ingredient that goes into obtaining this result is a simultaneous relative asymptotic orthogonality property, extending prior works in the literature. The proof uses technical tools including our uniform-flattening strategy for commutants in ultrapowers of II$_1$ factors.