On Kazhdan's property (T) and completeness of automorphism groups of Coxeter groups

关于Coxeter群自同构群的Kazhdan性质（T）和完备性

• 批准号: 438625854
• 负责人: Dr. Olga Varghese
• 金额: --
• 依托单位: Lehrstuhl für Algebra und Zahlentheorie
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/438625854?language=en
• 关键词: Kazhdan; property; completeness; automorphism; groups

This project is located in the area of geometric group theory. The basic principle of geometric group theory is to investigate algebraic properties of groups using geometric and topological methods. In this project I focus on groups which are defined via edge-labeled graphs, Coxeter groups, and their automorphism groups. One important property of a group is property (T). It was defined by Kazhdan for locally compact groups in terms of unitary representations. This property was reformulated in many different mathematical areas, in particular in geometric group theory. The first goal of this project is to show that the automorphism group of an infinite Coxeter group $Aut(W_\Gamma)$ does not have property (T) and give algebraic reasons that prohibit this group from having property (T). Many of the groups in geometric group theory are rigid, in the sense that their outer automorphism groups are finite. The second goal of this project is to characterize those automorphism groups of Coxeter groups in terms of $\Gamma$ which are complete (i.e. $Aut(Aut(W_\Gamma))=Inn(Aut(W_\Gamma)))$. Along the way I plan to investigate further properties of the automorphism group of a Coxeter group.

该项目位于几何群论领域。几何群论的基本原理是用几何和拓扑的方法研究群的代数性质。在这个项目中，我专注于通过边标记图，Coxeter群和它们的自同构群定义的群。群的一个重要性质是性质（T）。它是由Kazhdan定义的局部紧群的酉表示。这个性质在许多不同的数学领域被重新表述，特别是在几何群论中。本项目的第一个目标是证明无限Coxeter群$Aut（W_\Gamma）$的自同构群不具有性质（T），并给出禁止此群具有性质（T）的代数理由。几何群论中的许多群是刚性的，在这个意义上，它们的外自同构群是有限的。本项目的第二个目标是用$\Gamma$刻画Coxeter群的自同构群，即$Aut（Aut（W_\Gamma））=Inn（Aut（W_\Gamma））$。沿着的方式，我计划进一步研究的性质的自同构群的考克斯特群。