Property $t$ for Groups Graded by Root Systems (Memoirs of the American Mathematical Society) - Softcover

Ershov, Mikhail; Jaikan-zapirain, Andrei; Kassabov, Martin

Softcover

ISBN 10: 1470426048 ISBN 13: 9781470426040

Publisher: Amer Mathematical Society, 2017

Synopsis

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 $mathrm StPhi(R)$ and the elementary Chevalley group $mathbb EPhi(R)$ 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.

About this title

About the Author

Mikhail Ershov, University of Virginia, Charlottesville, Virginia.Andrei Jaikin-Zapirain, Universidad Autonoma de Madrid, Spain and Instituto de Ciencias Matematicas, Madrid, Spain.Martin Kassabov, Cornell University, Ithaca, New York, and University of Southampton, United Kingdom.

Edition details

• Publisher: Amer Mathematical Society
• Publication date: 2017
• Language: English
• ISBN 10: 1470426048
• ISBN 13: 9781470426040
• Binding: Paperback
• Number of pages: 135