Order separability of HNN-extensions and free products with commutative subgroups

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

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.