Bounded Generation of Submonoids of Heisenberg Groups

TL;DR

This paper proves that finitely generated submonoids of certain nilpotent groups are boundedly generated, and uses this to show the submonoid membership problem is decidable for a class of nilpotent groups.

Contribution

It establishes bounded generation of submonoids in nilpotent groups with specific commutator subgroup properties and applies this to decision problems.

Findings

Finitely generated submonoids are boundedly generated in certain nilpotent groups.

Decidability of submonoid membership problem for groups with specific commutator subgroup structure.

Reduction techniques connect algebraic properties to algorithmic decidability.

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 $2$.