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OverviewThe 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. Full Product DetailsAuthor: Mikhail Ershov , Andrei Jaikin-Zapirain , Martin KassabovPublisher: American Mathematical Society Imprint: American Mathematical Society Weight: 0.240kg ISBN: 9781470426040ISBN 10: 1470426048 Pages: 135 Publication Date: 30 October 2017 Audience: Professional and scholarly , Professional & Vocational Format: Paperback Publisher's Status: Active Availability: Temporarily unavailable  The supplier advises that this item is temporarily unavailable. It will be ordered for you and placed on backorder. Once it does come back in stock, we will ship it out to you. Table of ContentsIntroduction Preliminaries Generalized spectral criterion Root Systems Property $(T)$ for groups graded by root systems Reductions of root systems Steinberg groups over commutative rings Twisted Steinberg groups Application: Mother group with property $(T)$ Estimating relative Kazhdan constants Appendix A. Relative property $(T)$ for $({\rm St}_n(R)\ltimes R^n,R^n)$ Bibliography Index.ReviewsAuthor InformationMikhail Ershov, University of Virginia, Charlottesville, Virginia. Andrei Jaikin-Zapirain, Universidad Autonoma de Madrid, Spain and Instituto de Ciencias Matematicas, Madrid, Spain. Martin Kassabov, Cornell University, Ithaca, New York, and University of Southampton, United Kingdom. Tab Content 6Author Website:Countries AvailableAll regions |
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