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arXiv · 2409.04629

Critical groups in harmonic abelian quotients

Mariia Vetluzhskikh·Dmitry Zakharov
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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 $p:\widetilde{X}\to X$.

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Mariia Vetluzhskikh, Dmitry Zakharov. 2024-09-06. Critical groups in harmonic abelian quotients. https://arxiv.org/abs/2409.04629

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