## Groups Graded by Roots Sytems and Property T

#### Martin KassabovCornell University

We introduce and study the class of groups graded by root systems. We 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 we 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 $(T)$.

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