Hyperbolic groups and free constructions

Abstract

It is proved that the property of a group to be hyperbolic is preserved under HHN-extensions and amalgamated free products provided the associated (amalgamated) subgroups satisfy certain conditions. Some more general results about the preservation of hyperbolicity under graph products are also obtained. Using these results we describe the Q \mathbf {Q} -completion ( Q (\mathbf {Q} is the field of rationals) G Q G^{\mathbf {Q}} of a torsion-free hyperbolic group G G as a union of an effective chain of hyperbolic subgroups, and solve the conjugacy problem in G Q G^{\mathbf {Q}} .