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

TL;DR

This paper explores special graph covers where homology is not generated by primitive lifts, providing the smallest known examples with degree as low as 128, and offers an algorithm to identify suitable p-groups.

Contribution

It introduces the first known low-degree examples of graph covers with non-primitive homology and develops a character table-based algorithm for identifying relevant p-groups.

Findings

Examples of covers with degree as low as 128 exhibit non-primitive homology.

A character table-based algorithm can determine if a p-group admits such a cover.

Complete census of relevant p-groups under 1000 in order and rank.

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 $\pi_1(X)$. 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.