Pro-forests and nearly free profinite groups.
A group \(G\) is free if and only if there is a tree \(X\) on which \(G\) acts so that: \(G\) acts freely on \(X\), that is, no element of \(G\) except the identity has fixed points on \(X\); and \(G\) acts without inversions, that is, given any orientation of the tree, no element of \(G\) reverses an edge [see \textit{J.-P. Serre}, ``Arbres, amalgames, \(\mathrm{SL}_2\)'', Astérisque 46 (1977; Zbl 0369.20013)]. The goal of the paper under review is to extend this result to profinite groups. The main theorem states that a profinite group is nearly free, that is, it contains a dense abstract free group, if and only if it acts profreely on a protree. We refer to the paper for the details of the definitions of these terms.
scientific article; zbMATH DE number 5903697