Iwasawa Theory of Jacobians of Graphs

TL;DR

This paper explores the Iwasawa theory of Jacobians of graphs, focusing on derived graph towers and providing formulas for Sylow p-subgroups using classical invariants, bridging graph theory and number theory.

Contribution

It introduces a novel application of Iwasawa theory to the study of Jacobians of derived graph towers, deriving formulas for Sylow p-subgroups in this context.

Findings

Formulas for Sylow p-subgroups in infinite voltage p-towers.

Connection between graph Jacobians and Iwasawa invariants.

Extension of classical Iwasawa theory to graph invariants.

Abstract

The Jacobian group (also known as the critical group or sandpile group) is an important invariant of a finite, connected graph $X$; it is a finite abelian group whose cardinality is equal to the number of spanning trees of $X$ (Kirchhoff's Matrix Tree Theorem). A specific type of covering graph, called a derived graph, that is constructed from a voltage graph with voltage group $G$ is the object of interest in this paper. Towers of derived graphs are studied by using aspects of classical Iwasawa Theory (from number theory). Formulas for the orders of the Sylow $p$-subgroups of Jacobians in an infinite voltage $p$-tower, for any prime $p$, are obtained in terms of classical $\mu$ and $\lambda$ invariants by using the decomposition of a finitely generated module over the Iwasawa Algebra.