A group is matricially finite (MF) if its left regular representation can be approximated by maps into finite-dimensional matrices. It is purely MF (PMF) if these approximations can be chosen as actual homomorphisms, not just approximate ones. It is purely finite field (PFF) if each homomorphism lands in a finite group.
The paper on new sources of purely finite matricial fields (arXiv: 2603.24502) proves that amalgamated free products preserve these properties under a natural condition: if G is MF and H is a separable subgroup (the intersection of finite-index subgroups) and K is residually finite and MF, then G ∗_H (H × K) is MF — provided one of G or K is exact.
The consequences cascade. Graph products of residually finite exact MF groups are MF. Three-manifold groups are PFF. Groups that virtually embed into right-angled Artin groups are PFF. This last point connects to geometry: PFF is the property used in Antoine Song's approach to minimal surfaces.
The through-claim: approximability by matrices is preserved by the hardest group-theoretic operation. Free products and amalgams are the operations that most threaten finite-dimensionality — they create infinite groups from finite ones. The theorem says that if the pieces are well-approximated by matrices, so is their combination. Matricial finiteness is more robust than it appears, and its robustness has geometric consequences.
2603.24502. Operator algebras / group theory / matricial fields / amalgamated free products / 3-manifold groups.