Average Numbers of Homomorphisms to Random Modules over Free Group Algebras

Let $F$ be a finitely generated free group, and let $L$ be a finitely presented $\mathbb{F}_q[F]$-module. We study the average number $Λ_L(n)$ of $\mathbb{F}_q[F]$-module homomorphisms from $L$ to an $\mathbb{F}_q[F]$-module of dimension $n$ over $\mathbb{F}_q$. We show that, for all sufficiently large $n$, the quantity $Λ_L(n)$ is given by a rational function of $q^n$ and satisfies \[ Λ_L(n) = q^{χ(L)n} + \sum_{N\in A(L)} q^{χ(L/N)n}\bigl(1+O(q^{-n})\bigr), \] where $χ(L)$ denotes the Euler characteristic of $L$, and $A(L)$ is the set of nonzero $\mathbb{F}_q[F]$-submodules $N$ of $L$ that have no nonzero free quotients. Our proof is based on a theory of partial modules that may be of independent interest and have further applications.