Hyperbolic groups and free constructions

Abstract

We 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 $\mathbf{Q}$-completion of a torsion-free hyperbolic group has solvable word problem and conjugacy problem.

LaTex file, 44 pages. To appear in Trans. Am. Math. Soc