Rips construction and Kazhdan property (T)

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.

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