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

DuBose, Sage; Vallières, Daniel

Algebraic Combinatorics, Tome 6 (2023) no. 5, pp. 1331-1346

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.

Classification : 05C25, 11R18, 11R23, 11Z05
Keywords: Ihara zeta functions, Iwasawa theory, spanning trees

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