A constructive version of the Ribes-Zalesskiĭ product theorem. (Q1779874)

Let \(F\) be a free group and \(H_1,H_2,\dots,H_n\) is a collection of finitely generated subgroups of \(F\). It was proved by \textit{L. Ribes} and the reviewer [in Bull. Lond. Math. Soc. 25, No. 1, 37-43 (1993; Zbl 0811.20026)], that the product \(H_1H_2\cdots H_n\) is separable, i.e. closed in the profinite topology of \(F\). This means that for any element \(g\in F\setminus H_1H_2\cdots H_n\) there exists an epimorphism \(f\colon F\to K\) to a finite group \(K\) such that \(f(g)\not\in f(H_1H_2\cdots H_n)\). The main result of the paper is the explicit construction of the epimorphism \(f\). If \(H_1,\dots,H_n\) are closed in the pro-\(p\) topology, the methods of the paper also give an effective procedure to construct such an epimorphism \(f\) onto finite \(p\)-group \(K\). As a byproduct the authors prove the closedness in the pro-\(\mathbf H\) topology of the product \(H_1\cdots H_n\) of closed subgroups in the pro-\(\mathbf H\) topology of \(F\), where \(\mathbf H\) is a variety of groups satisfying the following condition: if \(G\in{\mathbf H}\) then there exists a cyclic group \(C\neq 1\) such that the wreath product \(C\wr G\in{\mathbf H}\).