Groups, Geometry, and Dynamics

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Volume 2, Issue 1, 2008, pp. 1–12
DOI: 10.4171/GGD/29

Published online: 2008-03-31

Rips construction and Kazhdan property (T)

Igor Belegradek[1] and Denis Osin[2]

(1) Georgia Institute of Technology, Atlanta, United States
(2) Vanderbilt University, Nashville, United States

We show that for any non-elementary hyperbolic group H and any finitely presented group Q, there exists a short exact sequence 1 → NGQ → 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.

Keywords: Hyperbolic group, Rips construction, Kazhdan property (T)

Belegradek Igor, Osin Denis: Rips construction and Kazhdan property (T). Groups Geom. Dyn. 2 (2008), 1-12. doi: 10.4171/GGD/29