Conjugacy separability in amalgamated products
In this chapter one studies how conjugacy separability in abstract groups is preserved under the formation of certain free products with amalgamation. The main result shows that one can construct conjugacy separable groups by forming a free product amalgamating a cyclic subgroup of groups which are either finitely generated free-by-finite or polycyclic-by-finite; in fact one can iterate this process to obtain new conjugacy separable groups; in particular residually finite groups. In addition to conjugacy separability one considers in this chapter a whole array of other properties that are preserved by constructing amalgamated free products of abstract groups with cyclic amalgamation, if one makes certain basic assumptions on the factors of the amalgamated free product. The main tools in most results in this chapter are related to the action of certain abstract groups on abstract trees and the action of certain profinite groups on profinite trees, and their inter-connections. In most cases in this chapter the pertinent groups are amalgamated free products and their profinite completions, and the pertinent trees and profinite trees are those canonically associated with amalgamated free products.
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Ribes, L. (2017). Conjugacy separability in amalgamated products. In Ergebnisse der Mathematik und ihrer Grenzgebiete. doi:10.1007/978-3-319-61199-0_15