Sheaves on Graphs and a Proof of the Hanna Neumann Conjecture

Note: Here is a remarkable and much shorter proof of the Hanna Neumann Conjecture, which simplifies, generalizes, and clarifies aspects of my sheaf theoretic proof, by Warren Dicks, written entirely in the language of skew group rings. remarkable simplifications, by Warren Dicks. You may want to read Dicks' version before mine...

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  • Abstract:

    The main goal of this paper is to prove the Hanna Neumann Conjecture; in fact, we prove a strengthened form of the conjecture. We study these conjectures using what we have called ``sheaves on graphs'' in \cite{friedman_sheaves}. We show that both conjectures are implied by the vanishing of a certain invariant, the ``maximum excess,'' of certain sheaves that we call $\rho$-kernels. Our approach involves ``graph Galois theory,'' an analogue of classical Galois theory in the graph setting. We use it to construct the $\rho$-kernels. We use the symmetry in Galois theory to argue that if the Strengthened Hanna Neumann Conjecture is false, then the maximum excess of ``most of'' these $\rho$-kernels must be large. We then give an inductive argument to show that this is impossible.