If F is a free abstract group, its profinite topology is the coarsest topology making F into a topological group, such that every group homomorphism from F into a finite group is continuous. It was shown by M. Hall Jr that every finitely generated subgroup of F is closed in that topology. Let H1, H2, …, Hn be finitely generated subgroups of F. J.-E. Pin and C. Reutenauer have conjectured that the product H1 H2 … Hn is a closed set in the profinite topology of F; also, they have shown that this conjecture implies a conjecture of J. Rhodes on finite semigroups. In this paper we give a positive answer to the conjecture of Pin and Reutenauer. Our method is based on the theory of profinite groups acting on graphs.
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Ribes et al. (1993) studied this question.