Free Products of Pro- C $\mathcal{C}$ Groups

Abstract

This is a central chapter in this book, and its results are frequently used throughout. The first section of the chapter contains a description a ‘sheaf of pro-\(\mathcal{C}\) groups’. Using this one defines a pro-\(\mathcal{C}\) group which is the free pro-\(\mathcal{C}\) product of the (fibers of the) sheaf. For a more internal viewpoint, one introduces the concept of ‘a collection of subgroups of a pro-\(\mathcal{C}\) group continuously indexed by a topological space (a profinite space)’: a prime example arises when one considers the stabilizers of a profinite group that acts on a profinite space. This allows us to describe when a pro-\(\mathcal{C}\) group is the free pro-\(\mathcal{C}\) product of some of its closed subgroups. After establishing the equivalence between the two viewpoints, external and internal, the chapter contains a large collection of basic properties of free products of pro-\(\mathcal{C}\) groups.