A norm for the cohomology of 2-complexes

We introduce a norm on the real 1-cohomology of finite 2-complexes determined by the Euler characteristics of graphs on these complexes. We also introduce twisted Alexander-Fox polynomials of groups and show that they give rise to norms on the real 1-cohomology of groups. Our main theorem states that for a finite 2-complex X, the norm on H^1(X; R) determined by graphs on X majorates the Alexander-Fox norms derived from \pi_1(X).