Construction of graph coverings with prescribed Iwasawa invariants

TL;DR

This paper explores the realization of Iwasawa invariants in graph coverings, demonstrating that any pair of invariants can be achieved through unramified or ramified $bZ_p$-coverings under certain conditions.

Contribution

It establishes the conditions under which arbitrary Iwasawa invariants can be realized in graph coverings, extending previous understanding to include ramified cases.

Findings

Any pair $(7, 8)$ with odd 7 can be realized in unramified coverings.

Any pair $(7, 8)$ can be realized in ramified coverings without parity restrictions.

The growth of spanning trees follows an analogue of Iwasawa's class number formula.

Abstract

For a $\mathbb{Z}_p$-covering of connected graphs, an analogue of Iwasawa's class number formula describes the growth of the number of spanning trees in terms of Iwasawa $\lambda$- and $\mu$-invariants. In this paper, we show that any pair $(\lambda, \mu)$ can be realized as the Iwasawa invariants of an unramified $\mathbb{Z}_p$-covering of a bouquet, provided that the necessary condition that $\lambda$ is odd is satisfied. We further show that any pair $(\lambda, \mu)$, without a parity condition, can be realized if we allow ramified $\mathbb{Z}_p$-coverings.