Invariant boundary distributions for finite graphs

Let Γ be the fundamental group of a finite connected graph G . Let M be an abelian group. A distribution on the boundary ∂Δ of the universal covering tree Δ is an M -valued measure defined on clopen sets. If M has no χ ( G ) -torsion, then the group of Γ-invariant distributions on ∂Δ is isomorphic to H 1 ( G , M ) .