Mixed-identity-freeness and primitivity of group rings

Abstract

We show that every countable group $G$ that is mixed-identity-free (MIF) and contains a non-abelian free subgroup has the following property: the group ring $KG$ is primitive for any field $K$. We also present a purely dynamical criterion that implies this result. Our criterion recovers several of the existing results on primitivity, including those involving acylindrically hyperbolic groups. Furthermore, our criterion also applies (positively) to a plethora of new examples, such as Thompson-like groups, commensurator groups of hyperbolic groups, some Kac-Moody groups, and many more.