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Boundary representations of non-positively curved groups

非正弯曲群的边界表示

基本信息

批准号：
EP/V002899/1
负责人：
Jan Spakula
金额：
$ 46.88万
依托单位：
University of Southampton
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2021
资助国家：
英国
起止时间：
2021 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FV002899%2F1
关键词：
Boundary representations non positively curved

项目摘要

The proposed research generally fits into the framework of Noncommutative Geometry, in particular research related to the Baum--Connes conjecture, analysis on groups, and representation theory. The Baum--Connes conjecture connects geometry, topology and algebra. From one point of view, it proposes a way to understand the algebraic topology (K-theory) of (a part of) the representation space of a group. While it is possible to effectively describe all the representations of (semisimple) Lie groups, this task is impossible for discrete groups in general. Here we propose to construct explicit families of representations for large classes of discrete groups, using geometry (non-positive curvature) and boundaries. They directly address important questions (Shalom's conjecture), relate to existing approaches to the Baum--Connes conjecture, and harmonic analysis on discrete groups. The proposed pathway combines ideas from analytic and geometric group theory, representation theory of Lie groups and random walks.The philosophy of this project is to capitalise on, and further develop, connections between Geometric Group Theory and Analysis/Noncommutative Geometry. We propose to construct a "compact picture" for (uniformly bounded) representations of prominent classes of non-positively curved groups.First, we deal with the case where one can do ``combinatorial harmonic analysis'', i.e. the case of groups acting properly on (finite dimensional) CAT(0) cube complexes.Second, we distill the main features of the construction and perform it with hyperbolic groups, thus establishing Shalom's conjecture.

所提出的研究一般适合于非交换几何的框架，特别是与Baum-Connes猜想，群分析和表示论相关的研究。Baum-Connes猜想连接了几何、拓扑和代数。从一个角度来看，它提出了一种理解群的表示空间（的一部分）的代数拓扑（K-理论）的方法。虽然可以有效地描述（半单）李群的所有表示，但这个任务对于一般的离散群是不可能的。在这里，我们建议构建明确的家庭表示的大类离散群体，使用几何（非正曲率）和边界。他们直接解决重要的问题（沙洛姆猜想），涉及到现有的方法鲍姆-康纳斯猜想，调和分析离散群体。建议的途径结合了分析和几何群论，李群和随机游动的表示理论的思想。这个项目的哲学是利用，并进一步发展，几何群论和分析/非交换几何之间的联系。我们建议建立一个“紧凑的图片”，（一致有界）表示的突出类的非正曲群。首先，我们处理的情况下，人们可以做“组合调和分析”，即情况下的群体适当地作用于（有限维）CAT（0）立方复形。其次，我们提取了该构造的主要特征，并用双曲群进行了构造，这就建立了沙洛姆的猜想。