Minasyan, Ashot On conjugacy separability of fibre products. (English) Zbl 1434.20012 Proc. Lond. Math. Soc. (3) 115, No. 6, 1170-1206 (2017). Summary: 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. Cited in 2 Documents MSC: 20E26 Residual properties and generalizations; residually finite groups 20F67 Hyperbolic groups and nonpositively curved groups PDFBibTeX XMLCite \textit{A. Minasyan}, Proc. Lond. Math. Soc. (3) 115, No. 6, 1170--1206 (2017; Zbl 1434.20012) Full Text: DOI arXiv