Property (T) for groups graded by root systems
Abstract
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 at least 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 at least 2 and a finitely generated commutative ring R with 1, the Steinberg group St_{\Phi}(R) and the elementary Chevalley group 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 at least 2, thereby providing a "unified" proof of expansion in these groups.
Keywords
Cite
@article{arxiv.1102.0031,
title = {Property (T) for groups graded by root systems},
author = {Mikhail Ershov and Andrei Jaikin-Zapirain and Martin Kassabov},
journal= {arXiv preprint arXiv:1102.0031},
year = {2014}
}
Comments
v2: 119 pages. Two new sections added (Section 9 and Appendix A); major revisions in Sections 5 and 8. In Section 9 it is proved that there exists a group with property (T) which surjects onto any finite simple group of Lie type and rank at least 2. Appendix A is based on the material from arXiv:math/0502237