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The SQ-universality and residual properties of relatively hyperbolic groups. (English) Zbl 1132.20022
The authors study residual properties of relatively hyperbolic groups. The group \(G\) is called SQ-universal if every countable group can be embedded into a quotient of \(G\). Let the group \(G\) be non-elementary and properly relatively hyperbolic (shortly PRH) with respect to the collection of subgroups \(\{H_\lambda\}_{\lambda\in\Lambda}\). It is shown that for each finitely generated group \(R\) there exists a quotient group \(Q\) of \(G\) and an embedding \(R\hookrightarrow Q\) such that \(Q\) is properly relatively hyperbolic with respect to the collection \([\psi(H_\lambda)]_{\lambda\in\Lambda}\cup R\), and \(H_\lambda\cap\ker(\psi)=H_\lambda\cap E_G(G)\) for each \(\lambda\in\Lambda\). Here \(\psi\colon G\to Q\) denotes the natural epimorphism and \(E_G(G)\) the maximal finite normal subgroup of \(G\). Consequently, any non-elementary PRH group is SQ-universal. Another main result: any two finitely generated non elementary PRH groups \(G_1,G_2\) have a common non-elementary PRH quotient.

MSC:
20F67 Hyperbolic groups and nonpositively curved groups
20E26 Residual properties and generalizations; residually finite groups
20E07 Subgroup theorems; subgroup growth
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