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Preliminaries Read chapter Read first chapter

2017 | OriginalPaper | Chapter

6. Graphs of Pro-\(\mathcal{C}\) Groups

Author : Luis Ribes

Published in: Profinite Graphs and Groups

Publisher: Springer International Publishing

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Abstract

This chapter contains a complete treatment of graphs of profinite groups \((\mathcal{G}, \varGamma)\) over profinite graphs \(\varGamma\) (this is a certain way of associating pro-\(\mathcal{C}\) groups \(\mathcal{G} (m)\) to vertices and edges \(m\) of \(\varGamma\)), their fundamental pro-\(\mathcal{C}\) groups \(\varPi(\mathcal{G}, \varGamma)\) and their standard (or universal covering) profinite graphs \(S(\mathcal{G}, \varGamma)\). It is proved that the standard graph of a graph of pro-\(\mathcal{C}\) groups is a \(\mathcal{C}\)-simply connected profinite graph. There are many examples dealing with the special cases of free pro-\(\mathcal{C}\) products, amalgamated products of profinite groups, HNN extensions, etc.
For applications to properties in abstract groups, in this chapter there is a study of the connections between a graph of abstract groups \((\mathcal{G}, \varGamma)\) over a finite graph \(\varGamma\) and a corresponding graph \((\bar{\mathcal{G}}, \varGamma)\) of profinite completions \(\bar{\mathcal{G}}(m)\), for every \(m\in \varGamma\). In some cases one can show that \(\varPi(\bar{\mathcal{G}}, \varGamma)\) is a profinite completion of the abstract fundamental group \(\varPi^{\mathrm{abs}}(\mathcal{G}, \varGamma)\) of \((\mathcal{G}, \varGamma)\), and that the universal covering tree \(S^{\mathrm{abs}}(\mathcal{G}, \varGamma)\) is densely embedded in \(S(\mathcal{G}, \varGamma)\). This is the case, for example, when dealing with free products of abstract residually finite groups, for graphs of finite groups or for certain types of amalgamated products, and then these connections can be used fruitfully in the study of some properties of the abstract fundamental groups \(\varPi^{\mathrm{abs}}(\mathcal{G}, \varGamma)\).

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previous chapter Free Products of Pro- Groups
next chapter Subgroups of Fundamental Groups of Graphs of Groups
Footnotes
1
In Serre (1980), Sect. I.5.3, Serre refers to the corresponding concept in the context of abstract groups, as a ‘universal covering’ graph relative to a graph of groups. In the present book, we occasionally use the expression ‘standard graph’ of a graph of groups, even when referring to abstract groups; the meaning should be clear by the context.
 
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Metadata
Title
Graphs of Pro- Groups
Author
Luis Ribes
Copyright Year
2017
Book
Profinite Graphs and Groups

Print ISBN: 978-3-319-61041-2

Electronic ISBN: 978-3-319-61199-0

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-3-319-61199-0_6

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