Invariant boundary distributions for finite graphs

TL;DR

This paper characterizes invariant boundary distributions for the fundamental group of a finite graph, linking them to the first homology group of the graph with coefficients in an abelian group.

Contribution

It establishes an isomorphism between invariant distributions on the boundary of the universal cover and the first homology group, under certain torsion conditions.

Findings

Invariant distributions correspond to elements of H_1(G, M).

The isomorphism holds when M has no torsion related to the graph's Euler characteristic.

Provides a homological classification of boundary measures for graph groups.

Abstract

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