Construction of graph coverings with prescribed Iwasawa invariants

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.