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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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