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Property $(T)$ for Groups Graded by Root Systems
About this Title
Mikhail Ershov, University of Virginia, Andrei Jaikin-Zapirain, Departamento de Matemáticas Universidad Autónoma de Madrid – and – Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM and Martin Kassabov, Cornell University and University of Southampton
Publication: Memoirs of the American Mathematical Society
Publication Year:
2017; Volume 249, Number 1186
ISBNs: 978-1-4704-2604-0 (print); 978-1-4704-4139-5 (online)
DOI: https://doi.org/10.1090/memo/1186
Published electronically: August 9, 2017
Keywords: Property $(T)$,
gradings by root systems,
Steinberg groups,
Chevalley groups
MSC: Primary 22D10, 17B22; Secondary 17B70, 20E42
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminaries
- 3. Generalized spectral criterion
- 4. Root Systems
- 5. Property $(T)$ for groups graded by root systems
- 6. Reductions of root systems
- 7. Steinberg groups over commutative rings
- 8. Twisted Steinberg groups
- 9. Application: Mother group with property $(T)$
- 10. Estimating relative Kazhdan constants
- A. Relative property $(T)$ for $({\mathrm {St}}_n(R)\ltimes R^n,R^n)$
Abstract
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 $(T)$. We 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.- A. Bak, The stable structure of quadratic modules, Thesis, Columbia University, 1969.
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