Rips construction and Kazhdan property (T)

2006/05/19 by Belegradek, Igor, Osin, Denis · 3 citations
#20F28 #20F65 #20F67 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.math/0605553

Abstract

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.

Cited by

• Undecidability of epimorphisms onto products of hyperbolic groups
• Remarks on approximability and stability for groups
• Relative solidity results and their applications to computations of some II₁ factor invariants

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