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Erschienen in:
2017 | OriginalPaper | Buchkapitel
5. Free Products of Pro-\(\mathcal{C}\) Groups
verfasst von : Luis Ribes
Erschienen in: Profinite Graphs and Groups
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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.
The case when all the factors in the free product are isomorphic to each other (corresponding to ‘constant sheaves’) is studied separately.
One section of the chapter explores the relationship between the topological weight of a profinite group \(G\) and the weight of a profinite space on which it acts, under appropriate conditions.