arXiv 1102.00312011-02math.GRmath.FAmath.RT

按根系分次群的（T）性质

Property (T) for groups graded by root systems

Mikhail Ershov, Andrei Jaikin-Zapirain, Martin Kassabov

备注：: 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

摘要：

引入并研究了按根系分次的群类。证明了：若Φ是秩至少为2的不可约经典根系，G是按Φ分次的群，则在一定的分次自然条件下，其根子群的并是G的Kazhdan子集.作为这个定理的主要应用，我们证明了对任意秩至少为2的既约不可约经典根系Φ和一个含1的非生成交换环R，Steinberg群St_Φ（R）和初等Chevalley群E_Φ（R）具有性质（T）.我们还证明了存在一个具有性质（T）的群，它映射到所有秩至少为2的李型有限单群上，从而提供了这些群中展开式的“统一”证明.

英文摘要：

We introduce and study the class of groups graded by root systems. We prove that if Φ is an irreducible classical root system of rank at least 2 and G is a group graded by Φ, 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 Φ of rank at least 2 and a finitely generated commutative ring R with 1, the Steinberg group St_Φ(R) and the elementary Chevalley group E_Φ(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.

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