Finitely-presented groups with long lower central series (Q1178340)

Let \(G\) be a group generated by elements \(x\), \(y\), \(z\), \(t\) subject to the following defining relations \(xy=yx\), \(txt^{-1}=x^{-1}\), \(tyt^{-1}=y^{-1}\), \(z=x[t,z]\). Denote by \(G_ n\) the \(n\)-th member of the lower central series in \(G\), and by \(G_ \omega\) the intersection of all groups \(G_ n\), \(n\geq 1\). Then \([G_ \omega,G]\neq G_ \omega\). The proof is based on the following remark. Let \(H\) be a group generated by elements \(x\), \(y\), \(t\) subject to the defining relations \(xy=yx\), \(txt^{-1}=x^{-1}\), \(tyt^{-1}=y^{-1}\). Consider in an algebraic closure \(\widehat H\) subgroups \(\widehat H_ n\), \(\widehat H_ \omega\) as above. Then \([\widehat H_ \omega,\widehat H]\neq\widehat H_ \omega\).