Sections of Submonoids of Nilpotent Groups
Doron Shafrir
TL;DR
The paper addresses the decidability of membership for products of finitely generated submonoids in finitely generated groups by embedding these products as sections in larger nilpotent-structured groups. It introduces constructions that place $A_1\cdots A_n$ as a section of a finitely generated submonoid inside $G \times H_{5}(\mathbb{Z})$, enabling a reduction of membership tests to submonoid membership in the product group; two proofs are presented, one using conjugate submonoids and another yielding a stronger result with $H=H_{5}(\mathbb{Z})$. These results yield new, simpler proofs of undecidability results (e.g., Romankov’s undecidable submonoid in $\mathcal{N}_2$) and establish a tight converse to Bodart’s reduction by connecting the problems of submonoid membership and product membership via the Hirsch length of the commutator subgroup. The approach leverages geometric plane constructions in the Heisenberg group and reductions to uniform problems in the product group, thereby clarifying the boundary between decidability and undecidability for these algebraic decision problems in nilpotent settings.
Abstract
We show that every product of f.g.\ submonoids of a group $G$ is a section of a f.g.\ submonoid of $G{\times}H_5(\mathbb{Z})$, where $H_5(\mathbb{Z})$ is a Heisenberg group. This gives us a converse of a reduction of Bodart, and a new simple proof of the existence of a submonoid of a nilpotent group of class 2 with undecidable membership problem.