A countable group G is said to be a matricial field (MF) if it admits a "strongly converging'' sequence of approximate homomorphisms into matrices, i.e, norms of polynomials converge to the corresponding value 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 a purely finite field (PFF) if the image of each homomorphism is finite. These questions have phenomenal applications to the study of C* and von Neumann algebras, spectral geometry, random walks and random graphs, spectral gaps of hyperbolic manifolds, minimal surface theory and Yau's conjectures, and even applied mathematics including but not limited to signal processing.
By developing a new operator algebraic approach to the MF 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 G is separable (i.e., H = \cap_{i \in \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 RAAGs. Prior to our paper, PFF results remained unknown even in the simple subcase of free products. These results are of further significance since PFF is the property that is used in Antoine Song's approach towards the existence of minimal surfaces in spheres of negative curvature.
