Equations in virtually class 2 nilpotent groups

TL;DR

This paper presents an algorithm to determine solvability of equations in virtually class 2 nilpotent groups with virtually cyclic commutator subgroups, extending previous results to finite extensions.

Contribution

It generalizes existing algorithms for class 2 nilpotent groups to include virtually extensions, broadening the class of groups where equation solvability can be decided.

Findings

Algorithm successfully decides equation solvability in the specified groups.

Extends previous work to include finite extensions of class 2 nilpotent groups.

Applicable to groups like the Heisenberg group with virtually cyclic commutator subgroups.

Abstract

We give an algorithm that decides whether a single equation in a group that is virtually a class $2$ nilpotent group with a virtually cyclic commutator subgroup, such as the Heisenberg group, admits a solution. This generalises the work of Duchin, Liang and Shapiro to finite extensions.