The First Roe Homology Group of Locally Finite Graphs

Abstract

We give a decomposition of the first group of so-called "Roe" homology of locally finite, connected graphs. We show that this group can be decomposed as a direct sum of two terms: the first counts the number of ends of the graph, while the second measures the existence of cycles that are not decomposable into smaller cycles (in some suitably coarse sense).