The first uniformly finite homology group with coefficients in $\mathbb{Z}$ and a characterisation of its vanishing in the transitive case

We study the first uniformly finite homology group of Block and Weinberger for uniformly locally finite graphs, with coefficients in $\mathbb{Z}$ and ${\mathbb{Z}}_2$. When the graph is a tree, or coefficients are in ${\mathbb{Z}}_2$, a characterisation of the group is obtained. In the general case, we describe three phenomena that entail non-vanishing of the group; their disjunction is shown to also be necessary for non-vanishing in the case of transitive graphs.