Conjugacy in free products and in free-by-finite groups
Let 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 C -separable’ if for any pair of elements x, y∈ R, these elements are conjugate in R if and only if their images in every finite quotient of R which is in C are conjugate (there is an analogous property of ‘subgroup conjugacy C -separability’, if one replaces elements with finitely generated subgroups). A subgroup H of R is said to be ‘conjugacy C -distinguished’ if whenever y∈ 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∈ C of R. In Chap. 14 it is shown that the properties of conjugacy C -separability and subgroup conjugacy C -separability are preserved by taking free products of abstract groups. It is also shown that an abstract free-by- C group (an extension of a free abstract group by a group in C) is both conjugacy C -separable and subgroup conjugacy C -separable; in these groups every finitely generated pro- C closed subgroup is conjugacy 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.
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Ribes, L. (2017). Conjugacy in free products and in free-by-finite groups. In Ergebnisse der Mathematik und ihrer Grenzgebiete. doi:10.1007/978-3-319-61199-0_14