On ${\mathbb{Z}}_{\ell }^d$-towers of graphs

Abstract

Let $\ell$ be a rational prime. We show that an analogue of a conjecture of Greenberg in graph theory holds true. More precisely, we show that when $n$ is sufficiently large, the $\ell$-adic valuation of the number of spanning trees at the $n$th layer of a ${\mathbb{Z}}_{\ell }^d$-tower of graphs is given by a polynomial in ${\ell }^n$ and $n$ with rational coefficients of total degree at most $d$ and of degree in $n$ at most one.