Property (T) for groups graded by root systems
著者
書誌事項
Property (T) for groups graded by root systems
（Memoirs of the American Mathematical Society, no. 1186）
American Mathematical Society, 2017
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注記
"Volume 249, number 1186 (seventh of 8 numbers), September 2017"
Bibliography: p. 133-134
Includes index
内容説明・目次
内容説明
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 ${\mathrm St}_{\Phi}(R)$ 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.
目次
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.
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