2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/153042We investigate which free constructions (amalgamated products and HNN-extensions) over word hyperbolic groups produce groups that are again word hyperbolic. A complete answer is obtained for the case when the amalgamated subgroups are virtually cyclic. The results are applied, in particular, to show that a ${\bf Q}$-completion of a torsion-free hyperbolic group has solvable word problem and conjugacy problem.LaTex file, 44 pages. To appear in Trans. Am. Math. SocGroup Theorytext