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 ELn(R), 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.