Submonoid Membership in n-dimensional lamplighter groups and S-unit equations
<|reference_start|>Submonoid Membership in n-dimensional lamplighter groups and S-unit equations: 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.<|reference_end|>
@article{dong2024submonoid,
    title={Submonoid Membership in n-dimensional lamplighter groups and S-unit
  equations},
    author={Ruiwen Dong},
    journal={arXiv preprint arXiv:2409.07077},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.07077},
    primaryClass={math.GR cs.FL math.NT}
}