Invariant boundary distributions for finite graphs

Let G be a finite connected graph, and let ∆ be its universal covering tree. The edges of ∆ are directed and each geometric edge corresponds to two directed edges δ and δ. Let ∆ be the set of vertices and ∆ the set of directed edges of ∆. The boundary ∂∆ is the set of equivalence classes of infinite semi-geodesics in ∆, where two semi-geodesics are equivalent if they agree except on finitely many edges. The boundary has a natural compact totally disconnected topology. The fundamental group Γ of G is a free group which acts on ∆ and on ∂∆. If M is an abelian group then an M-valued distribution on ∂∆ is a finitely additive M-valued measure μ defined on the clopen subsets of ∂∆. By integration, a distribution may be regarded as an M-linear function on the group C(∂∆,M) of locally constant M-valued functions on ∂∆. Let D(∂∆,M) be the additive group of all Γ-invariant M-valued distributions on ∂∆ and let