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

Abstract

The well-known theorem of Shalom–Vaserstein and Ershov–Jaikin-Zapirain states that the group ${\mathrm{EL}<!--FUNCTION\; APPLICATION-->}_n(\mathcal{ℛ})$, generated by elementary matrices over a finitely generated commutative ring $\mathcal{ℛ}$, has Kazhdan’s property (T) as soon as $n\ge 3$. This is no longer true if the ring $\mathcal{ℛ}$ is replaced by a commutative rng (a ring but without the identity) due to nilpotent quotients ${\mathrm{EL}<!--FUNCTION\; APPLICATION-->}_n(\mathcal{ℛ}∕{\mathcal{ℛ}}^k)$. We prove that even in such a case the group ${\mathrm{EL}<!--FUNCTION\; APPLICATION-->}_n(\mathcal{ℛ})$ satisfies a certain property that can substitute property (T), provided that $n$ is large enough.