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.
1
Georgia Institute of Technology, Atlanta, United States
2
Vanderbilt University, Nashville, United States
Igor Belegradek; Denis Osin. Rips construction and Kazhdan property (T). Groups, geometry, and dynamics, Tome 2 (2008) no. 1, pp. 1-12. doi: 10.4171/ggd/29
@article{10_4171_ggd_29,
author = {Igor Belegradek and Denis Osin},
title = {Rips construction and {Kazhdan} property {(T)}},
journal = {Groups, geometry, and dynamics},
pages = {1--12},
year = {2008},
volume = {2},
number = {1},
doi = {10.4171/ggd/29},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/29/}
}
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