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

2024-09-11

arXiv.org

AI Summary

This paper investigates the decidability of the submonoid membership problem for the $n$-dimensional lamplighter group $(mathbb{Z}/pmathbb{Z}) wr mathbb{Z}^n$ and more general semidirect products. The core method reduces the problem to solving $S$-unit equations over $mathbb{Z}^n$-modules, leveraging tools from algebraic group theory, module theory over $mathbb{Z}^n$, $p$-automatic sequences, and reductions from the knapsack problem. Three main contributions are established: (i) the first explicit construction of a group admitting a finite-index subgroup with decidable submonoid membership, while the ambient group has an undecidable instance; (ii) the first effective proof that solution sets of $S$-unit equations over $mathbb{Z}^n$-modules are $p$-automatic; and (iii) the decidability of the submonoid membership problem in $(mathbb{Z}/pmathbb{Z}) wr mathbb{Z}^n$, together with the effective $p$-automaticity of its knapsack solution set.

Technology Category

Constraint Satisfaction and Optimization: Satisfiability Modulo TheoriesKnowledge Representation and Reasoning: Computational Complexity of ReasoningReasoning under Uncertainty: Sequential Decision Making

Application Category

Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphsEconomics, Online Markets and Human Computation: Cost models of using LLMs in production systemsSemantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

We show that Submonoid Membership is decidable in n-dimensional lamplighter groups $(mathbb{Z}/pmathbb{Z}) wr mathbb{Z}^n$ for any prime $p$ and integer $n$. More generally, we show decidability of Submonoid Membership in semidirect products of the form $mathcal{Y} times mathbb{Z}^n$, 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 $mathcal{Y} times mathbb{Z}^n$ 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 $mathcal{Y} times mathbb{Z}^n$ is effectively $p$-automatic.

Problem

• Deciding Submonoid Membership in n-dimensional lamplighter groups
• Solving S-unit equations over Laurent polynomial ring modules
• Determining decidable Submonoid Membership in semidirect products

Innovation

• Decides Submonoid Membership in lamplighter groups
• Reduces problem to S-unit equations
• Uses effectively p-automatic solution sets