Residual finiteness, QCERF, and fillings of hyperbolic groups

TL;DR

This paper explores the implications of residual finiteness in hyperbolic groups, demonstrating that it leads to the separability of quasi-convex subgroups through the use of relatively hyperbolic Dehn filling techniques.

Contribution

It establishes a conditional link between residual finiteness of hyperbolic groups and the separability of their quasi-convex subgroups, advancing understanding of hyperbolic group properties.

Findings

If all hyperbolic groups are residually finite, then all their quasi-convex subgroups are separable.

Utilizes relatively hyperbolic Dehn filling as a key tool in the proof.

Provides a new approach to understanding subgroup separability in hyperbolic groups.

Abstract

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.