Finite approximation and commutators in free groups. (Q1887466)

Let \(F=F_m\) be a free group of rank \(m\) with the set \(A=\{a_1,\dots,a_m\}\) of free generators, and let \(w(x_1,\dots,x_n,a_1,\dots,a_m)=1\) be an equation over \(F\). A class \(\mathcal G\) of such equations is said to be finitely approximable when each equation from \(\mathcal G\) has a solution in \(F\) if and only if it has a solution in every finite quotient of \(F\). It is well known that every equation of the form \(x^{-1}gx=f\) (``conjugacy''), or of the form \(x^k=f\) (``power''), where \(g,f \in F\), is finitely approximable. It was shown by the author and T. Coulbois, answering a question by G. Sabbagh, that in general there exist not finitely approximable equations. The aim of the paper under review is to prove that the equation \([x_1,x_2]=g\), \(g\in F\), is finitely approximable. In other words, the author proves that the set of commutators of \(F\) is closed for the profinite topology.