  We show that for any non--elementary hyperbolic group $H$ and any finitely
presented group $Q$, there exists a short exact sequence $1\to N\to G\to Q\to
1$, 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 $x^n=1$ 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.
