Critical Groups of Graphs with Dihedral Actions II

TL;DR

This paper studies the structure of critical groups of finite connected graphs with dihedral symmetries, extending previous work by decomposing these groups via quotients related to the automorphism subgroup actions.

Contribution

It introduces a novel decomposition of the critical group for graphs with dihedral automorphisms, paralleling a theorem for algebraic curves with similar symmetries.

Findings

Critical group decomposes into critical groups of quotient graphs.

Extension of previous work on harmonic dihedral actions.

Analogous to Kani and Rosen's theorem for algebraic curves.

Abstract

In this paper we consider the critical group of finite connected graphs which admit harmonic actions by the dihedral group $D_n$, extending earlier work by the author and Criel Merino. In particular, we show that the critical group of such a graph can be decomposed in terms of the critical groups of the quotients of the graph by certain subgroups of the automorphism group. This is analogous to a theorem of Kani and Rosen which decomposes the Jacobians of algebraic curves with a $D_n$-action.