Property (T) for noncommutative universal lattices

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≥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 (Dymara and Januszkiewicz in Invent. Math. 150(3):579–627, 2002).