Groups graded by root systems and property (T).

Ershov, Mikhail; Jaikin-Zapirain, Andrei; Kassabov, Martin; Zhang, Zezhou · Proc Natl Acad Sci U S A · 2014

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Abstract

We establish property (T) for a large class of groups graded by root systems, including elementary Chevalley groups and Steinberg groups of rank at least 2 over finitely generated commutative rings with 1. We also construct a group with property (T) which surjects onto all finite simple groups of Lie type and rank at least two.