Semigroup intersection problems in the Heisenberg groups

TL;DR

This paper proves that the problem of determining whether finitely generated sub-semigroups in Heisenberg groups and certain nilpotent groups intersect is computationally feasible, providing polynomial-time algorithms for these cases.

Contribution

It establishes PTIME decidability for Intersection Emptiness in Heisenberg and 2-step nilpotent groups, and decidability of Orbit Intersection in the Heisenberg group $ ext{H}_3( ext{Q})$, extending prior results.

Findings

Intersection Emptiness is PTIME decidable in Heisenberg groups over algebraic number fields.

Intersection Emptiness is decidable in direct products of Heisenberg groups.

Orbit Intersection is decidable within the Heisenberg group $ ext{H}_3( ext{Q})$.

Abstract

We consider two algorithmic problems concerning sub-semigroups of Heisenberg groups and, more generally, two-step nilpotent groups. The first problem is Intersection Emptiness, which asks whether a finite number of given finitely generated semigroups have empty intersection. This problem was first studied by Markov in the 1940s. We show that Intersection Emptiness is PTIME decidable in the Heisenberg groups $\operatorname{H}_{n}(\mathbb{K})$ over any algebraic number field $\mathbb{K}$, as well as in direct products of Heisenberg groups. We also extend our decidability result to arbitrary finitely generated 2-step nilpotent groups. The second problem is Orbit Intersection, which asks whether the orbits of two matrices under multiplication by two semigroups intersect with each other. This problem was first studied by Babai et al. (1996), who showed its decidability within commutative matrix groups. We show that Orbit Intersection is decidable within the Heisenberg group $\operatorname{H}_{3}(\mathbb{Q})$.