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

TL;DR

This paper introduces a new property that can replace Kazhdan's property (T) for certain groups generated by elementary matrices over rings without identity, especially when traditional property (T) fails.

Contribution

The authors establish a substitute property for groups generated by elementary matrices over commutative rngs, extending the scope beyond rings with identity.

Findings

The new property holds for large enough n.

It applies to groups over rings without identity.

It generalizes Kazhdan's property (T) in non-lattice contexts.

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 $n\geq3$. 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.