Groups graded by root systems and property (T)

Significance In 1967 Kazhdan invented certain property of locally compact groups defined in terms of their unitary representations, called property (T). It has important applications in representation theory, theory of algebraic groups, ergodic theory, geometric group theory, operator algebras, network theory, and expander graphs. While a variety of methods for constructing groups with property (T) have been developed, the question of whether a given group has property (T) remains open in many important cases. In this paper we develop a method that allows to establish property (T) for a large class of groups, which includes elementary Chevalley groups and Steinberg groups of rank at least 2 over finitely generated commutative rings with 1. This opens a door for further interesting applications. 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.