  We establish a new spectral criterion for Kazhdan's property $(T)$ which is
applicable to a large class of discrete groups defined by generators and
relations. As the main application, we prove property $(T)$ for the groups
$EL_n(R)$, where $n\geq 3$ and $R$ is an arbitrary finitely generated
associative ring. We also strengthen some of the results on property $(T)$ for
Kac-Moody groups from a paper of Dymara and Januszkiewicz (Invent. Math 150
(2002)).
