Equations in nilpotent groups

TL;DR

This paper presents an algorithm to decide single equations in the Heisenberg group and similar groups, but proves the undecidability of systems of equations in broader non-abelian free nilpotent groups.

Contribution

It introduces a decision algorithm for single equations in certain nilpotent groups and establishes undecidability results for systems of equations in others.

Findings

Algorithm exists for single equations in Heisenberg and related groups

Decision problem for systems of equations is unsolvable in non-abelian free nilpotent groups

Applies to all two-step nilpotent groups with rank-one commutator

Abstract

We show that there exists an algorithm to decide any single equation in the Heisenberg group in finite time. The method works for all two-step nilpotent groups with rank-one commutator, which includes the higher Heisenberg groups. We also prove that the decision problem for systems of equations is unsolvable in all non-abelian free nilpotent groups.