A note on the continuous extensions of injective morphisms between free groups to relatively free profinite groups.
Let \(\varphi\colon F(A)\to F(B)\) be an embedding of free groups of finite ranks. Then \(\varphi\) induces a continuous epimorphism \(\widehat\varphi\colon\widehat F_{\mathbf V}(A)\to\widehat F_{\mathbf V}(B)\) of the pro-\(\mathbf V\) completions, where \(\mathbf V\) is a pseudovariety of finite groups. The paper is concerned with the injectivity of \(\widehat\varphi\). The main result is the following Theorem. Let \(\mathbf V\) be a pseudovariety of finite groups such that free groups are residually \(\mathbf V\) and let \(H=\varphi(F(A))\). Then the following conditions are equivalent: \(\bullet\) \(\widehat\varphi\) is injective. \(\bullet\) The pro-\(\mathbf V\) topology of \(H\) coincides with the topology induced by the pro-\(\mathbf V\) topology of \(F(B)\). -- If, in addition, \(\mathbf V\) is extension closed, these properties are equivalent to: \(\bullet\) \(H\) and the closure \(\text{Cl}(H)\) of \(H\) in \(F(B)\) have the same rank.
scientific article; zbMATH DE number 2109795