Skip to Main Content

AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution

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)
Published electronically: August 9, 2017
Keywords: Property $(T)$, gradings by root systems, Steinberg groups, Chevalley groups
MSC: Primary 22D10, 17B22; Secondary 17B70, 20E42

View full volume PDF

View other years and numbers:

Table of Contents


  • 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 $({\mathrm {St}}_n(R)\ltimes R^n,R^n)$


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.

References [Enhancements On Off] (What's this?)