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Erschienen in:
2017 | OriginalPaper | Buchkapitel
6. Graphs of Pro-\(\mathcal{C}\) Groups
verfasst von : Luis Ribes
Erschienen in: Profinite Graphs and Groups
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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 \(\varPi^{\mathrm{abs}}(\mathcal{G}, \varGamma)\).