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We show that for any non--elementary hyperbolic group $H$ and any finitely presented group $Q$, there exists a short exact sequence $1to Nto Gto Qto 1$, 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 $x^n=1$ 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.
We construct several series of explicit presentations of infinite hyperbolic groups enjoying Kazhdans property (T). Some of them are significantly shorter than the previously known shortest examples. Moreover, we show that some of those hyperbolic Ka
Let R be a finitely generated commutative ring with 1, let A be an indecomposable 2-spherical generalized Cartan matrix of size at least 2 and M=M(A) the largest absolute value of a non-diagonal entry of A. We prove that there exists an integer n=n(A
We show that the automorphism group of a graph product of finite groups $Aut(G_Gamma)$ has Kazhdans property (T) if and only if $Gamma$ is a complete graph.
We show that for a large class $mathcal{W}$ of Coxeter groups the following holds: Given a group $W_Gamma$ in $mathcal{W}$, the automorphism group ${rm Aut}(W_Gamma)$ virtually surjects onto $W_Gamma$. In particular, the group ${rm Aut}(G_Gamma)$ is
We prove the existence of a close connection between spaces with measured walls and median metric spaces. We then relate properties (T) and Haagerup (a-T-menability) to actions on median spaces and on spaces with measured walls. This allows us to exp