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# memo_has_moved_text();Property $(T)$ for Groups Graded by Root Systems

Mikhail Ershov, Andrei Jaikin-Zapirain and Martin Kassabov

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

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Chapters

• Chapter 1. Introduction
• Chapter 2. Preliminaries
• Chapter 3. Generalized spectral criterion
• Chapter 4. Root Systems
• Chapter 5. Property $(T)$ for groups graded by root systems
• Chapter 6. Reductions of root systems
• Chapter 7. Steinberg groups over commutative rings
• Chapter 8. Twisted Steinberg groups
• Chapter 9. Application: Mother group with property $(T)$
• Chapter 10. Estimating relative Kazhdan constants
• Appendix 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 is an irreducible classical root system of rank and is a group graded by , then under certain natural conditions on the grading, the union of the root subgroups is a Kazhdan subset of . As the main application of this theorem we prove that for any reduced irreducible classical root system of rank and a finitely generated commutative ring with , the Steinberg group and the elementary Chevalley group have property . We also show that there exists a group with property which maps onto all finite simple groups of Lie type and rank , thereby providing a âĂĲunifiedâĂİ proof of expansion in these groups.