Invariant boundary distributions for finite graphs

Abstract

Let $Γ$ be the fundamental group of a finite connected graph $\mathcal G$. Let $\mathfrak M$ be an abelian group. A {\it distribution} on the boundary $\partialΔ$ of the universal covering tree $Δ$ is an $\mathfrak M$-valued measure defined on clopen sets. If $\mathfrak M$ has no $χ(\mathcal G)$-torsion then the group of $Γ$-invariant distributions on $\partialΔ$ is isomorphic to $H_1(\mathcal G,\mathfrak M)$.