Property (T)

财产（T）

• 批准号: 441426599
• 负责人: Professor Dr. Roman Sauer
• 金额: --
• 依托单位: Arbeitsgruppe Topologie und Geometrische Gruppentheorie
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2020
• 资助国家: 德国
• 起止时间: 2019-12-31 至 2023-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/441426599?language=en
• 关键词: Property

The goal of the project is to obtain new insights into the ways in which groups of key importance in Geometric Group Theory, namely Chevalley groups, Mapping Class Groups, and Out(F_n), act on Hilbert spaces by isometries. More specifically, we will investigate the presence of Kazhdan's property (T), and we will provide lower bounds for Kazhdan constants; studying both the qualitative and quantitative aspects of the problem will allow us to paint a detailed picture of the rigidity landscape for the above groups.Our main goal is to prove that Mapping Class Groups have property (T). This would solve a long-standing open problem which has acquired some notoriety in the Geometric Group Theory community. We intend to do this by combining combinatorial and representation-theoretic methods with the power of modern computers, similarly to the way we established property (T) for Out(F_n).Our secondary objectives are: to obtain a new, unified proof of property (T) for Chevalley groups over the integers (which will also compute lower bounds for Kazhdan constants); to remove the computer-assisted aspects of the current proof of property (T) for Out(F_n); and to combine the previous two points in order to prove property (T) for Chevalley groups over a large variety of rings.

该项目的目标是获得对几何群论中至关重要的群（即 Chevalley 群、映射类群和 Out(F_n)）通过等距作用于希尔伯特空间的方式的新见解。更具体地说，我们将调查 Kazhdan 属性 (T) 的存在，并提供 Kazhdan 常数的下界；研究问题的定性和定量方面将使我们能够详细描绘上述组的刚性景观。我们的主要目标是证明映射类组具有属性 (T)。这将解决一个长期悬而未决的问题，该问题在几何群论界已经获得了一些恶名。我们打算通过将组合和表示论方法与现代计算机的能力相结合来实现这一点，类似于我们为 Out(F_n) 建立属性 (T) 的方式。我们的次要目标是：获得整数上 Chevalley 群的新的、统一的属性 (T) 证明（这也将计算 Kazhdan 常数的下界）；删除 Out(F_n) 的当前财产证明 (T) 的计算机辅助方面；并结合前两点来证明Chevalley群在各种环上的性质（T）。