Graph coverings and (im)primitive homology: some new examples of exceptionally low degree

Abstract

Given a finite covering of graphs $f : Y \to X$, it is not always the case that $H_1(Y;\mathbb{C})$ is spanned by lifts of primitive elements of $π_1(X)$. In this paper, we study graphs for which this is not the case, and we give here the simplest known nontrivial examples of covers with this property, with covering degree as small as 128. Our first step is focusing our attention on the special class of graph covers where the deck group is a finite $p$-group. For such covers, there is a representation-theoretic criterion for identifying deck groups for which there exist covers with the property. We present an algorithm for determining if a finite $p$-group satisfies this criterion that uses only the character table of the group. Finally, we provide a complete census of all finite $p$-groups of rank $\geq 3$ and order $< 1000$ satisfying this criterion, all of which are new examples.