Property ($T$) for Groups Graded by Root Systems

ISBN-13: 9781470426040 / Angielski

Mikhail Ershov;Andrei Jaikin-Zapirain;Martin Kassabov

Opis

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 ${mathrm St}_(R)$ and the elementary Chevalley group $mathbb E_(R)$ 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.

Spis treści

• Introduction
• 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.