The residual finiteness of (hyperbolic) automorphism-induced HNN-extensions

TL;DR

This paper classifies when automorphism-induced HNN-extensions are residually finite, offering a way to construct non-residually finite examples, but shows these methods cannot produce counterexamples to Gromov's conjecture on hyperbolic groups.

Contribution

It provides a classification criterion for residual finiteness of automorphism-induced HNN-extensions based on subgroup separability, and demonstrates limitations in constructing counterexamples to Gromov's conjecture.

Findings

Classification criterion for residual finiteness

Construction method for non-residually finite extensions

Limitations on counterexamples to Gromov's conjecture

Abstract

We classify finitely generated, residually finite automorphism-induced HNN-extensions in terms of the residual separability of a single associated subgroup. This classification provides a method to construct automorphism-induced HNN-extensions which are not residually finite. We prove that this method can never yield a "new" counter-example to Gromov's conjecture on the residual finiteness of hyperbolic groups.