Submonoid Membership in n-dimensional lamplighter groups and S-unit equations

TL;DR

This paper proves that Submonoid Membership is decidable in certain lamplighter groups and related structures, using reductions to S-unit equations and automata theory, revealing new decidability boundaries in group theory.

Contribution

It establishes the first decidability results for Submonoid Membership in specific lamplighter groups and modules, connecting group theory with S-unit equations and automata.

Findings

Decidability of Submonoid Membership in lamplighter groups and modules.

Solution sets of S-unit equations are effectively p-automatic.

Knapsack Problem solutions are effectively p-automatic.

Abstract

We show that Submonoid Membership is decidable in n-dimensional lamplighter groups $(\mathbb{Z}/p\mathbb{Z}) \wr \mathbb{Z}^n$ for any prime $p$ and integer $n$. More generally, we show decidability of Submonoid Membership in semidirect products of the form $\mathcal{Y} \rtimes \mathbb{Z}^n$, where $\mathcal{Y}$ is any finitely presented module over the Laurent polynomial ring $\mathbb{F}_p[X_1^{\pm}, \ldots, X_n^{\pm}]$. Combined with a result of Shafrir (2024), this gives the first example of a group $G$ and a finite index subgroup $\widetilde{G} \leq G$, such that Submonoid Membership is decidable in $\widetilde{G}$ but undecidable in $G$. To obtain our decidability result, we reduce Submonoid Membership in $\mathcal{Y} \rtimes \mathbb{Z}^n$ to solving S-unit equations over $\mathbb{F}_p[X_1^{\pm}, \ldots, X_n^{\pm}]$-modules. We show that the solution set of such equations is effectively $p$-automatic, extending a result of Adamczewski and Bell (2012). As an intermediate result, we also obtain that the solution set of the Knapsack Problem in $\mathcal{Y} \rtimes \mathbb{Z}^n$ is effectively $p$-automatic.