Equations in virtually class 2 nilpotent groups

Abstract

We construct an algorithm to decide if in a class 2 nilpotent group an equation ω = 1 that contains a variable X such that the exponent sum of X within ω is non-zero admits a solution. Besides the existence of such an X in ω, there are no restrictions on any other variables. We do this by associating the equation to a system of integer equations and congruences equivalent to it, and give an algorithm to solve this system. We also construct an algorithm to decide if any equation in a group that is virtually the Heisenberg group admits a solution.