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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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