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

TL;DR

This work establishes the decidability of Submonoid Membership in all $n$-dimensional lamplighter groups $(\mathbb{Z}/p\mathbb{Z}) \wr \mathbb{Z}^n$ for any prime $p$ and $n$, and more generally for semidirect products $\mathcal{Y} \rtimes \mathbb{Z}^n$ with $\mathcal{Y}$ a finitely presented $\mathbb{F}_p[X_1^{\pm}, \ldots, X_n^{\pm}]$-module. The authors reduce Submonoid Membership to solving S-unit equations over these modules and prove that solution sets are effectively $p$-automatic, also showing the Knapsack Problem is effectively $p$-automatic in this setting. A notable consequence, combined with Shafrir (2024), is the first example of a group $G$ with a finite-index subgroup $\widetilde{G} \le G$ such that Submonoid Membership is decidable in $\widetilde{G}$ but undecidable in $G$, highlighting that decidability can fail to ascend through finite extensions. The methodology intricately blends combinatorial group theory with effective commutative algebra and automata theory, leveraging Noether normalization, primary decomposition, and $p$-automaticity to achieve decidability results in metabelian groups.

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.

Theorems & Definitions (31)

• Lemma 2.1: baumslag1981computable
• Lemma 2.2: baumslag1981computable or baumslag1994algorithmic
• Lemma 2.3: wolper2000construction
• Theorem 3.1
• Corollary 3.2
• Theorem 3.3
• Theorem 3.4
• Proposition 4.1: potthast2020submonoid
• Example 4.2: label=exa:cont
• Remark 4.3