Rips construction and Kazhdan property (T)
by
Igor Belegradek
Georgia Tech
Coauthors: Denis Osin (CUNY)

We note that the small cancellation theory over hyperbolic groups yields an attractive version of the Rips construction: for each non-elementary hyperbolic group H and a finitely presented group Q there is a short exact sequence 1 -> N -> G -> Q -> 1 where N is a quotient of H, and G is hyperbolic. This also has a relatively hyperbolic version where H, G are relatively hyperbolic and Q is finitely generated. Sample applications:

1) Any finitely generated group embeds as a finite index subgroup in Out(N) where N has property (T) (or more generally a quotient of any given non-elementary relatively hyperbolic group). One can also arrange that Out(Aut(N))=1.

2) There exists a large torsion-free hyperbolic group G and an element g in G such that the group < G : g^n > is not large for all odd n.

3) There is a torsion-free hyperbolic group that is representation rigid, but not representation superrigid.

4) Property (T) is not recursively recognizable in the class of hyperbolic groups.

Date received: March 10, 2006