Probabilistic Hanna Neumann Conjectures

We develop a theory of polymatroids on Stallings core graphs, which provides a new technique for proving lower bounds on stable invariants of words and subgroups in free groups $F$, and for upper bounds on their probability for mapping, under a random homomorphism from $F$ to a finite group $G$, into some subgroup of $G$. As a result, we prove the gap conjecture on the stable $K$-primitivity rank by Ernst-West, Puder and Seidel, prove a conjecture of Reiter about the number of solutions to a system of equations in a finite group action, and give a unified proof of the "rank-1 Hanna Neumann conjecture" by Wise and its higher rank analogue. We further show that the stable compressed rank and its $q$-analogue coincide with the decay rate of many-words measure on stable actions of finite simple groups of large rank. Finally, we conjecture an analogue of the Hanna Neumann conjecture over fields, and suggest that every finite group action is associated to some version of the HNC.