Probabilistic Hanna Neumann Conjectures

TL;DR

This paper introduces a new theory of polymatroids on Stallings core graphs to establish bounds on subgroup invariants in free groups, proving several longstanding conjectures and proposing new analogues over fields.

Contribution

It develops a novel polymatroid framework on Stallings graphs, proving key conjectures in group theory and establishing connections to finite group actions and field analogues.

Findings

Proved the gap conjecture on stable K-primitivity rank.

Confirmed Reiter's conjecture on solutions to equations in finite groups.

Provided a unified proof of the rank-1 Hanna Neumann conjecture.

Abstract

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.