Bounded Generation of Submonoids of Heisenberg Groups

TL;DR

The paper investigates submonoid generation and membership in nilpotent groups, showing that when $h([G,G])=1$ every finitely generated submonoid is computably boundedly generated, enabling explicit decompositions into products of cyclic submonoids. This bounded-generation result, together with Malcev-coordinate techniques, yields a uniform reduction of the submonoid membership problem to knapsack-type problems, and via Bodart's framework extends decidability to nilpotent groups with $h([G,G])\le 2$. The work both generalizes knapsack decidability to broader nilpotent settings (including Heisenberg-type groups) and outlines a practical decision procedure by translating submonoid questions into finite-index subgroup problems with linear and quadratic constraints. It also documents open questions and avenues to tighten bounds and broaden applicability to additional group classes.

Abstract

If $G$ is a nilpotent group and $[G,G]$ has Hirsch length $1$, then every f.g. submonoid of $G$ is boundedly generated, i.e. a product of cyclic submonoids. Using a reduction of Bodart, this implies the decidability of the submonoid membership problem for nilpotent groups $G$ where $[G,G]$ has Hirsch length $2$.

Theorems & Definitions (18)

• Theorem 1
• Lemma 2
• proof
• Remark 1
• Lemma 3
• proof
• Lemma 4
• proof
• Proposition 5
• Theorem 6