On the finite approximability of equations in finitely generated nilpotent groups (Q1191217)

Let \(G\) be a finitely generated nilpotent group, \(a\) an arbitrary element of \(G\), and \(w\) a word in \(x_ 1,\dots,x_ n\) which does not belong to the commutator subgroup of the free group generated by \(x_ 1,\dots,x_ n\). The author proves that if \(w(x_ 1,\dots,x_ n) = a\) has no solution in \(G\), then there exists an epimorphism \(\varphi: G \to K\) onto a finite group \(K\) such that the equation \(w(x_ 1,\dots,x_ n) = \varphi(a)\) has no solution in \(K\) either. Based on this, an algorithm for deciding whether \(w(x_ 1,\dots,x_ n) = a\) can be solved in \(G\) is presented. On the other hand, a result of N. N. Repin is quoted, showing that no such algorithm exists without restrictions on the word \(w\).