On conjugacy separability of fibre products

TL;DR

This paper investigates conjugacy separability in subdirect products of free and hyperbolic groups, establishing criteria, constructing examples, and characterizing p-conjugacy separable subgroups, linking them to residual finiteness of p-groups.

Contribution

It provides necessary and sufficient conditions for conjugacy separability in fibre products and constructs examples illustrating the nuanced behavior of conjugacy separability in overgroups.

Findings

Existence of finitely presented groups with all finite index subgroups conjugacy separable but with non-conjugacy separable overgroups.

Construction of finitely presented groups with non-conjugacy separable subgroups of index 2 and conjugacy separable normal overgroups.

Characterization of p-conjugacy separable subdirect products and their relation to residually finite p-groups.

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 posses 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.