A substitute for Kazhdan’s property (T) for universal nonlattices

The well-known theorem of Shalom-Vaserstein and Ershov-Jaikin-Zapirain states that the group EL n (R), generated by elementary matrices over a finitely generated commutative ring R, has Kazhdan's property (T) as soon as n ≥ 3.This is no longer true if the ring R is replaced by a commutative rng (a ring but without the identity) due to nilpotent quotients EL n (R/R k ).We prove that even in such a case the group EL n (R) satisfies a certain property that can substitute property (T), provided that n is large enough.