Combinatorial iterated integrals and the harmonic volume of graphs

Let Γ be a connected bridgeless metric graph, and fix a point v of Γ. We define combinatorial iterated integrals on Γ along closed paths at v, a unipotent generalization of the usual cycle pairing and the combinatorial analogue of Chen's iterated integrals on Riemann surfaces. These descend to a bilinear pairing between the group algebra of the fundamental group of Γ at v and the tensor algebra on the first homology of Γ, ∫:Zπ1(Γ,v)×TH1(Γ,R)→R. We show that this pairing on the two-step unipotent quotient of the group algebra allows one to recover the base-point v up to well-understood finite ambiguity. We encode the data of this structure as the combinatorial harmonic volume which is valued in the tropical intermediate Jacobian. We also give a potential-theoretic characterization for hyperellipticity for graphs.