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.
Cite this article
Igor Belegradek, Denis Osin, Rips construction and Kazhdan property (T). Groups Geom. Dyn. 2 (2008), no. 1, pp. 1–12DOI 10.4171/GGD/29