Matchings and product growth in modular abelian independence groups

Abstract

We unify two matching theories, one for finite subsets of groups and the other for finite-dimensional subspaces in a field extension. To achieve this, we study groups equipped with a compatible finitary matroid structure, termed here independence groups. Applying Rado's independent transversal theorem, we derive necessary and sufficient rank criteria for matchability between finite-rank sets. In the setting of a modular abelian independence group $G$, we develop an analogue of the $e$-transform from additive number theory, derive structural matching criteria, and characterize a global matching property by the absence of a submonoid $H$ satisfying $1<\rho(H)<\rho(G)$ and $\rho(H)<\infty$, where $\rho$ denotes rank. Examples of modular abelian independence groups are given and examined in the matching context. Arising from this matching theory, but formulated without any reference to it, is a product-growth bound that generalizes the Cauchy--Davenport theorem: we define a parameter $\mu(G)$ and prove that $\rho(XY)\geq \min\{\mu(G),\rho(X)+\rho(Y)-1\}$ for all nonempty finite-rank subsets $X,Y$ of $G$. Furthermore, $\rho(XY)$ is shown to be controlled from below by a submonoid of $G$ that stabilizes a flat, a phenomenon reminiscent of Kneser's theorem.