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Residual finiteness, QCERF and fillings of hyperbolic groups. (English) Zbl 1229.20037
From the introduction: We prove that if every hyperbolic group is residually finite, then every quasi-convex subgroup of every hyperbolic group is separable. The main tool is relatively hyperbolic Dehn filling.
A group $$G$$ is residually finite (or RF) if for every $$g\in G\setminus\{1\}$$, there is some finite group $$F$$ and an epimorphism $$\varphi\colon G\to F$$ so that $$\varphi(g)\neq 1$$. In more sophisticated language $$G$$ is RF if and only if the trivial subgroup is closed in the profinite topology on $$G$$.
If $$H<G$$, then $$H$$ is separable if for every $$g\in G\setminus H$$, there is some finite group $$F$$ and an epimorphism $$\varphi\colon G\to F$$ so that $$\varphi(g)\not\in\varphi(H)$$. Equivalently, the subgroup $$H$$ is separable in $$G$$ if it is closed in the profinite topology on $$G$$.
If every finitely generated subgroup of $$G$$ is separable, $$G$$ is said to be LERF or subgroup separable. If $$G$$ is hyperbolic, and every quasi-convex subgroup of $$G$$ is separable, we say that $$G$$ is QCERF.
In this paper, we show that if every hyperbolic group is RF, then every hyperbolic group is QCERF. Theorem 0.1. If all hyperbolic groups are residually finite, then every quasi-convex subgroup of a hyperbolic group is separable.

##### MSC:
 20F67 Hyperbolic groups and nonpositively curved groups 20E26 Residual properties and generalizations; residually finite groups 20F65 Geometric group theory 57M07 Topological methods in group theory
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