Rational Subsets in HNN-Extensions and Amalgamated Products

of a group H, where A and B are isomorphic subgroups of H and φ : A → B is an isomorphism, results from adding to H a new generator t such that the conjugation of the subgroup A ≤ H by t realizes the isomorphism φ. We also say that A and B in (2) are the associated subgroups. One of the first important applications of HNN-extensions was a more transparent proof of the celebrated result of Novikov and Boone on the existence of a finitely presented group with an undecidable word problem, see e.g. [LS77]. Such a group can be constructed by a series of HNN-extensions starting from a free group. This shows that arbitrary HNN-extensions do not have good algorithmic properties. In