Critical groups in harmonic abelian quotients

TL;DR

This paper investigates the relationship between critical groups of graphs under harmonic covers, especially Galois abelian covers, by computing kernel orders and relating spanning trees through matroid theory.

Contribution

It provides explicit formulas for the kernel size of the pushforward map on Jacobians in harmonic abelian covers of graphs.

Findings

Kernel order of the pushforward map is computed for Galois abelian covers.

Relationship between spanning trees of the cover and base graph is established.

Connection to Zaslavsky's bias matroid is demonstrated.

Abstract

A harmonic cover of graphs $p:\widetilde{X}\to X$ induces a surjective pushforward morphism $p_*:\operatorname{Jac}(\widetilde{X})\to \operatorname{Jac}(X)$ on the critical groups. In the case when $p$ is Galois with abelian Galois group, we compute the order of the kernel of $p_*$, and hence the relationship between the numbers of spanning trees of $\widetilde{X}$ and $X$, in terms of Zaslavsky's bias matroid associated to the cover $p:\widetilde{X}\to X$.