Non Uniform Kazhdan Constant for Linear Groups

Abstract

A group $\Gamma$ generated by a finite set $S$ has Property $(T)$ if the associated Kazhdan constant $\kappa(\Gamma,S)$ is positive. If so, the same holds for every finite generating set $S$. There are Property $(T)$ groups which are uniformly $(T)$, and others which are not. It is a well-known problem whether the classical examples of Property $(T)$ groups, $\mathrm{SL}_n(\mathbb{Z})$, $n\geq 3$, are uniformly $(T)$ or not. We show that they are not. Moreover, the same holds for every infinite finitely generated linear group.