Equations in virtually class \(2\) nilpotent groups (Q6566799)

An equation with the variable set \(V\) in a group \(G\) has the form \(w=1\) for some element \(w \in G \ast F(V)\). The single equation problem in a group \(G\) is the decision question as to whether there is an algorithm for \(G\) that takes as input an equation \(w=1\) in \(G\) and outputs whether or not \(w=1\) admits a solution. In [\textit{M. Duchin} et al., Proc. Am. Math. Soc. 143, No. 11, 4723--4731 (2015; Zbl 1330.20047)], it is proven that the single equation problem in finitely generated nilpotent groups of class \(2\) with a virtually cyclic commutator subgroup is decidable. The aim of the paper under review is to generalize the previous result. The main theorem is the following:\N\NTheorem 1.1: The single equation problem in a group that is virtually nilpotent of class \(2\) with virtually cyclic commutator subgroup is decidable. (The reviewer reports that the previous statement in the paper is confusing due to a typographical error.)\N\NThe assumption that the commutator subgroup is virtually cyclic cannot be completely removed; \textit{V. A. Roman'kov} [J. Group Theory 19, No. 3, 497--514 (2016; Zbl 1361.20025)] gave an example of a finitely generated nilpotent group of class \(2\), where it is undecidable whether equations of the form \([X_{1},X_{2}]=g\), where \(g \in G\) is a constant, admit solutions.