Property $(T)$ for noncommutative universal lattices

TL;DR

This paper introduces a new spectral criterion for Kazhdan's property (T) and applies it to prove property (T) for noncommutative universal lattices, specifically for groups like $EL_n(R)$ with finitely generated rings.

Contribution

It develops a novel spectral criterion for property (T) and establishes it for a broad class of groups, including noncommutative universal lattices and certain Kac-Moody groups.

Findings

Proved property (T) for $EL_n(R)$ with $n\,geq 3$ and finitely generated rings.

Established a new spectral criterion applicable to groups defined by generators and relations.

Extended results on property (T) for Kac-Moody groups.

Abstract

We establish a new spectral criterion for Kazhdan's property $(T)$ which is applicable to a large class of discrete groups defined by generators and relations. As the main application, we prove property $(T)$ for the groups $EL_n(R)$, where $n\geq 3$ and $R$ is an arbitrary finitely generated associative ring. We also strengthen some of the results on property $(T)$ for Kac-Moody groups from a paper of Dymara and Januszkiewicz (Invent. Math 150 (2002)).