Invariant boundary distributions for finite graphs

Let $\mathrm{\Gamma }$ be the fundamental group of a finite connected graph $\mathcal G$\mathrm{.}Let$\mathfrak M$beanabeliangroup\mathrm{.}Adistributionontheboundary$\partial\Delta$oftheuniversalcoveringtree$\Delta$isan$\mathfrak M$-valuedmeasuredefinedonclopensets\mathrm{.}If$\mathfrak M$hasno$\chi(\mathcal G)$-torsionthenthegroupof$\Gamma$-invariant distributions on $\mathrm{\partial }\mathrm{\Delta }$ is isomorphic to $H_1(\mathcal{G},\mathfrak{M})$.