Real free groups and the absolute Galois group of \(R(t)\)
A profinite group \(G\) is called ``real projective'' if the set \(\text{Inv}(G)\) of involutions of \(G\) is a closed subset of \(G\), and to every epimorphism \(\alpha: B\to A\) of finite groups and every homomorphism \(\phi: G\to A\) such that for every \(\delta\in \text{Inv}(G)\setminus \text{Ker}(\phi)\) there exists an \(\varepsilon\in \text{Inv}(B)\) for which \(\alpha (\varepsilon)=\phi (\delta)\), there exists a homomorphism \(\gamma: G\to B\) such that \(\alpha \circ \gamma =\phi.\) The authors develop the notion of a ``real free'' profinite group in such a way that the real projective groups are exactly the closed subgroups of real free profinite groups. This parallels the fact that projective profinite groups are the closed subgroups of free profinite groups. A profinite group \(D\) is called ``real free'' if it contains disjoint closed subsets \(X\) and \(Y\) such that \(X\subset \text{Inv}(D)\), \(1\in Y\) and every continuous map \(\phi\) from \(X\cup Y\) into a profinite group \(G\), such that \(\phi (x)^ 2=1\) for every \(x\in X\) and \(\phi (1)=1\), extends to a unique homomorphism of \(D\) into \(G\). It is shown that the absolute Galois group of \(R(t)\) is real free for real closed fields \(R\).
scientific article; zbMATH DE number 3922802
scientific article