A substitute for Kazhdan's property (T) for universal non-lattices

Abstract

The well-known theorem of Shalom--Vaserstein and Ershov--Jaikin-Zapirain states that the group ${\mathrm{EL}}_n(\mathcal{R})$, 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 ${\mathrm{EL}}_n(\mathcal{R}/{\mathcal{R}}^k)$. In this paper, we prove that even in such a case the group ${\mathrm{EL}}_n(\mathcal{R})$ satisfies a certain property that can substitute property (T), provided that $n$ is large enough.

20 pages; fixed typo (v2); minor changes (v3)

Cited by in corpus (1)

• On the Ground State Energies of Discrete and Semiclassical Schrödinger Operators