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On conjugacy separability of fibre products. (English) Zbl 1434.20012

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.

MSC:

20E26 Residual properties and generalizations; residually finite groups
20F67 Hyperbolic groups and nonpositively curved groups
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