2 Igor Belegradek And

We show that for any non–elementary hyperbolic group H and any finitely presented group Q, there exists a short exact sequence 1 → N → G → Q → 1, where G is a hyperbolic group and N is a quotient group of H . As an application we construct a hyperbolic group that has the same n–dimensional complex representations as a given finitely generated group, show that adding relations of the form xn = 1 to a presentation of a hyperbolic group may drastically change the group even in case n >> 1, and prove that some properties (e.g. properties (T) and FA) are not recursively recognizable in the class of hyperbolic groups. A relatively hyperbolic version of this theorem is also used to generalize results of Ollivier–Wise on outer automorphism groups of Kazhdan groups.