Chip-firing on graphs of groups

TL;DR

This paper extends classical graph theory concepts like the Laplacian and Jacobian to graphs of groups, establishing key theorems and formulas, and analyzing group actions on these structures.

Contribution

It introduces the Laplacian and Jacobian for graphs of groups, proving analogues of classical theorems and defining pushforward and pullback maps under group actions.

Findings

Established the Laplacian and Jacobian for graphs of groups.

Proved analogues of the matrix tree theorem and class number formula.

Derived a formula for the kernel of the pushforward map when G=Z/2Z.

Abstract

We define the Laplacian matrix and the Jacobian group of a finite graph of groups. We prove analogues of the matrix tree theorem and the class number formula for the order of the Jacobian of a graph of groups. Given a group $G$ acting on a graph $X$, we define natural pushforward and pullback maps between the Jacobian groups of $X$ and the quotient graph of groups $X/\!/G$. For the case $G=\mathbb{Z}/2\mathbb{Z}$, we also prove a combinatorial formula for the order of the kernel of the pushforward map.