Property (T) for groups graded by root systems - Ershov, Mikhail, 1978-; Jaikin-Zapirain, Andrei; Kassabov, Martin, 1977- (Memoirs of the American Mathematical Society, 2017)
The authors introduce and study the class of groups graded by root systems. They prove that if Φ is an irreducible classical root system of rank ≥ 2 and G is a group graded by Φ, then under certain natural conditions on the grading, the union of the root subgroups is a Kazhdan subset of G. As the main application of this theorem the authors prove that for any reduced irreducible classical root system Φ of rank ≥ 2 and a finitely generated commutative ring R with 1, the Steinberg group {\mathrm St}_{Φ}(R) and the elementary Chevalley group \mathbb E_{Φ}(R) have property (T).