A new source of purely finite matricial fields

Abstract

A countable group $G$ is said to be \emph{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 \emph{purely MF} (PMF) if these maps are actual homomorphisms, and $G$ is further \emph{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, we list some below: 1. The Brown--Douglas--Fillmore semigroups of many new examples of reduced group $C^*$-algebras are shown to be not groups. 2. Arbitrary group doubles $G*_HG$ of MF (resp., PMF, PFF) over separable subgroups $H$ are MF (resp., PMF, PFF). Moreover, $G*H$ is PFF whenever $G,H$ are PFF, and either $G$ or $H$ is exact. 3. 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 M. Magee and J. Thomas. 4. The open problem of proving PFF for fundamental groups of closed hyperbolic 3-manifolds is resolved. This has geometric significance in the theory of minimal surfaces via A. Song's approach.