PHP decompositions and primitivity of group rings for linear groups

Abstract

We show that every countable group $G$ that has Ozawa's property $\mathrm{PHP}$ and contains a non-abelian free subgroup has the following property: the group ring $KG$ is primitive for any field $K$. As a consequence, we deduce that all non-trivial countable linear groups with trivial amenable radical have this property. Along the way, we prove that the class of groups with $\mathrm{PHP}$ enjoys some permanence properties that are of independent interest.

arXiv comment9 pages. Comments welcome!