Strong property (T) for higher rank lattices

IF 5.4 3区 材料科学 Q2 CHEMISTRY, PHYSICAL
M. Salle
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引用次数: 11

Abstract

We prove that every lattice in a product of higher rank simple Lie groups or higher rank simple algebraic groups over local fields has Vincent Lafforgue's strong property (T). Over non-archimedean local fields, we also prove that they have strong Banach proerty (T) with respect to all Banach spaces with nontrivial type, whereas in general we obtain such a result with additional hypotheses on the Banach spaces. The novelty is that we deal with non-cocompact lattices, such as $\mathrm{SL}_n(\Z)$ for $n \geq 3$. To do so, we introduce a stronger form of strong property (T) which allows us to deal with more general objects than group representations on Banach spaces that we call two-step representations, namely families indexed by a group of operators between different Banach spaces that we can compose only once. We prove that higher rank groups have this property and that this property passes to undistorted lattices.
高阶格的强性质(T)
证明了高阶单李群或高阶简单代数群在局部域上的积中的每一个格都具有Vincent Lafforgue的强性质(T)。在非阿基基德局部域上,我们还证明了它们对于所有非平凡类型的Banach空间都具有强Banach性质(T),而在一般情况下,我们在Banach空间上通过附加的假设得到了这样的结果。新奇之处在于我们处理的是非紧致格子,比如$n \geq 3$的$\mathrm{SL}_n(\Z)$。为此,我们引入了强性质(T)的一种更强的形式,它允许我们处理比巴拿赫空间上的群表示更一般的对象,我们称之为两步表示,即由不同巴拿赫空间之间的一组算子索引的族,我们只能组合一次。我们证明了高秩群具有这一性质,并且这一性质传递给了未扭曲的格。
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来源期刊
ACS Applied Energy Materials
ACS Applied Energy Materials Materials Science-Materials Chemistry
CiteScore
10.30
自引率
6.20%
发文量
1368
期刊介绍: ACS Applied Energy Materials is an interdisciplinary journal publishing original research covering all aspects of materials, engineering, chemistry, physics and biology relevant to energy conversion and storage. The journal is devoted to reports of new and original experimental and theoretical research of an applied nature that integrate knowledge in the areas of materials, engineering, physics, bioscience, and chemistry into important energy applications.
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