A new source of purely finite matricial fields

Abstract: A countable group $G$ is said to be matricial field (MF) if it admits a strongly converging sequence of approximate homomorphisms into matrices; i.e, the norms of polynomials converge to those in the left regular representation. $G$ is then said to be purely MF (PMF) if this sequence of maps into matrices can be chosen as actual homomorphisms. $G$ is further said to be purely finite field (PFF) if the image of each homomorphism is finite. By developing a new operator algebraic approach to these problems, we are able to prove the following result bringing several new examples into the fold. Suppose $G$ is a MF (resp., PMF, PFF) group and $H<G$ is separable (i.e., $H=\cap_{i\in \mathbb{N}}H_i$ where $H_i<G$ are finite index subgroups) and $K$ is a residually finite MF (resp., PMF, PFF) group. If either $G$ or $K$ is exact, then the amalgamated free product $G*_{H}(H\times K)$ is MF (resp., PMF, PFF). Our work has several applications. Firstly, as a consequence of MF, the Brown--Douglas--Fillmore semigroups of many new reduced $C^*$-algebras are not groups. Secondly, we obtain that arbitrary graph products of residually finite exact MF (resp., PMF, PFF) groups are MF (resp., PMF, PFF), yielding a significant generalization of the breakthrough work of Magee--Thomas. Thirdly, our work resolves the open problem of proving PFF for 3-manifold groups. More generally all groups that virtually embed into RAAGs are PFF. Prior to our work, PFF was not known even in the case of free products. Our results are of geometric significance since PFF is the property that is used in Antoine Song's approach in the theory of minimal surfaces.