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

Let $\mathrm{\ell }$ 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 $\mathrm{\ell }$-adic valuation of the number of spanning trees at the $n$th layer of a ${\mathbb{Z}}_{\mathrm{\ell }}^d$-tower of graphs is given by a polynomial in ${\mathrm{\ell }}^n$ and $n$ with rational coefficients of total degree at most $d$ and of degree in $n$ at most one.