On ℤ ℓ d -towers of graphs

Abstract

Let ℓ 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 ℓ-adic valuation of the number of spanning trees at the nth layer of a ℤ ℓ d -tower of graphs is given by a polynomial in ℓ n and n with rational coefficients of total degree at most d and of degree in n at most one.