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Description of  Property $t$ for Groups Graded by Root Systems (Memoirs of the American Mathematical Society)

1470426048 pdf 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 $mathrm StPhi(R)$ and the elementary Chevalley group $mathbb EPhi(R)$ 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.