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Published in:
2017 | OriginalPaper | Chapter
14. Conjugacy in Free Products and in Free-by-Finite Groups
Author : Luis Ribes
Published in: Profinite Graphs and Groups
Publisher: Springer International Publishing
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Abstract
Let \(\mathcal{C}\) be an extension-closed pseudovariety of finite groups (i.e, a class of finite groups closed under subgroups, quotients and extensions; e.g., the class of all finite groups). An abstract group \(R\) is ‘conjugacy \(\mathcal{C}\)-separable’ if for any pair of elements \(x,y\in R\), these elements are conjugate in \(R\) if and only if their images in every finite quotient of \(R\) which is in \(\mathcal{C}\) are conjugate (there is an analogous property of ‘subgroup conjugacy \(\mathcal{C}\)-separability’, if one replaces elements with finitely generated subgroups). A subgroup \(H\) of \(R\) is said to be ‘conjugacy \(\mathcal{C}\)-distinguished’ if whenever \(y\in R\), then \(y\) has a conjugate in \(H\) if and only if the same holds for the images of \(y\) and \(H\) in every quotient group \(R/N\in \mathcal{C}\) of \(R\).
In Chap. 14 it is shown that the properties of conjugacy \(\mathcal{C}\)-separability and subgroup conjugacy \(\mathcal{C}\)-separability are preserved by taking free products of abstract groups. It is also shown that an abstract free-by-\(\mathcal{C}\) group (an extension of a free abstract group by a group in \(\mathcal{C}\)) is both conjugacy \(\mathcal{C}\)-separable and subgroup conjugacy \(\mathcal{C}\)-separable; in these groups every finitely generated pro-\(\mathcal{C}\) closed subgroup is conjugacy \(\mathcal{C}\)-distinguished. The basic tools for proving these results are related to the study of minimal invariant subtrees developed in Chap. 8 for the actions of groups on trees.