Property ($T$) for Groups Graded by Root Systems

Introduces and studies the class of groups graded by root systems. The authors prove that if $/Phi$ is an irreducible classical root system of rank $/geq 2$ and $G$ is a group graded by $/Phi$, then under certain natural conditions on the grading, the union of the root subgroups is a Kazhdan subset of $G$.

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The authors introduce and study the class of groups graded by root systems. They prove that if $/Phi$ is an irreducible classical root system of rank $/geq 2$ and $G$ is a group graded by $/Phi$, 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 $/Phi$ of rank $/geq 2$ and a finitely generated commutative ring $R$ with $1$, the Steinberg group ${/mathrm St}_{/Phi}(R)$ and the elementary Chevalley group $/mathbb E_{/Phi}(R)$ have property $(T)$. They also show that there exists a group with property $(T)$ which maps onto all finite simple groups of Lie type and rank $/geq 2$, thereby providing a ``unified'' proof of expansion in these groups.