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Property (𝑇) for Groups Graded by Root Systems
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Property (𝑇) for Groups Graded by Root Systems
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eBook
Property (𝑇) for Groups Graded by Root Systems
2017
Overview
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.
Publisher
American Mathematical Society
Subject
ISBN
9781470426040, 1470426048
