Rips construction and Kazhdan property (T)
Igor Belegradek
Georgia Institute of Technology, Atlanta, United StatesDenis Osin
Vanderbilt University, Nashville, United States
Abstract
We show that for any non-elementary hyperbolic group and any finitely presented group , there exists a short exact sequence , where is a hyperbolic group and is a quotient group of . As an application we construct a hyperbolic group that has the same -dimensional complex representations as a given finitely generated group, show that adding relations of the form to a presentation of a hyperbolic group may drastically change the group even in case , 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. 1–12
DOI 10.4171/GGD/29