On conjugacy separability of fibre products. Issue 6 (18th August 2017)

• Record Type: Journal Article
• Authors: Minasyan, Ashot
• Is Part Of: Proceedings of the London Mathematical Society. Volume 115:Issue 6(2017)
• Page Start: 1170
• Page End: 1206
• Publication Date: 2017-08-18
• Subjects: 20E26 (primary) -- 20F67 (secondary); Mathematics -- Periodicals; Mathematics; Periodicals; 510
• DOI: 10.1112/plms.12065
• Languages: English
• ISSNs: 0024-6115

Abstract

Abstract: In this paper we study conjugacy separability of subdirect products of two free (or hyperbolic) groups. We establish necessary and sufficient criteria and apply them to fibre products to produce a finitely presented group G 1 in which all finite index subgroups are conjugacy separable, but which has an index 2 overgroup that is not conjugacy separable. Conversely, we construct a finitely presented group G 2 which has a non‐conjugacy separable subgroup of index 2 such that every finite index normal overgroup of G 2 is conjugacy separable. The normality of the overgroup is essential in the last example, as such a group G 2 will always possess an index 3 overgroup that is not conjugacy separable. Finally, we characterize p ‐conjugacy separable subdirect products of two free groups, where p is a prime. We show that fibre products provide a natural correspondence between residually finite p ‐groups and p ‐conjugacy separable subdirect products of two non‐abelian free groups. As a consequence, we deduce that the open question about the existence of an infinite finitely presented residually finite p ‐group is equivalent to the question about the existence of a finitely generated p ‐conjugacy separable full subdirect product of infinite index in the direct product of two free groups.