On the Iwasawa theory of Cayley graphs

TL;DR

This paper applies Iwasawa theory to Cayley graphs, analyzing algebraic invariants in graph towers and revealing connections between graph theory, number theory, and group theory.

Contribution

It introduces a novel graph-theoretic approach to Iwasawa invariants, including factorization of Iwasawa polynomials and conditions for invariant vanishing in Cayley graph towers.

Findings

Factorization of Iwasawa polynomials for Cayley graphs

Decomposition of μ- and λ-invariants in graph towers

Conditions for invariants to vanish in complete graphs

Abstract

This paper explores Iwasawa theory from a graph theoretic perspective, focusing on the algebraic and combinatorial properties of Cayley graphs. Using representation theory, we analyze Iwasawa-theoretic invariants within $\mathbb{Z}_\ell$-towers of Cayley graphs, revealing connections between graph theory, number theory, and group theory. Key results include the factorization of associated Iwasawa polynomials and the decomposition of $\mu$- and $\lambda$-invariants. Additionally, we apply these insights to complete graphs, establishing conditions under which these invariants vanish.