Strongly singular MASA's and mixing actions in finite von Neumann algebras

Abstract

Let $Γ$ be a countable group and let $Γ_0$ be an infinite abelian subgroup of $Γ$. We prove that if the pair $(Γ,Γ_0)$ satisfies some combinatorial condition called (SS), then the abelian subalgebra $A=L(Γ_0)$ is a singular MASA in $M=L(Γ)$ which satisfies a weakly mixing condition. If moreover it satisfies a stronger condition called (ST), then it provides a singular MASA with a strictly stronger mixing property. We describe families of examples of both types coming from free products, HNN extentions and semidirect products, and in particular we exhibit examples of singular MASA's that satisfy the weak mixing condition but not the strong mixing one.