Joel Friedman’s proof of the strengthened Hanna Neumann conjecture

1.1 Notation. As Bourbaki intended, we let N denote the set of finite cardinals, {0, 1, 2, . . .}. Throughout this section, let F be a field. We shall write dim(V ) to denote the F-dimension of an F-module V . Throughout this section, let (Z,VZ,EZ,EZ ι,τ −→ VZ) be a finite (oriented) graph; here, Z is a finite set, VZ ⊆ Z, EZ = Z −VZ, and ι and τ are functions. Each e ∈ EZ has an associated picture of the form ιe • e −→ • or ιe=τe • e . We let the symbol Z also denote the graph. We shall use the standard concepts of subgraph, connected graph, component of a graph, tree, tree component of a graph, and graph map. We write δ(Z) := |EZ|−|VZ| and r(Z) :=max{δ(Y ) : Y is a subgraph of Z}. Each subgraph Y of Z with δ(Y ) = r(Z) is called a δ-maximizer in Z. The intersection of all the δ-maximizers in Z is denoted supercore(Z).