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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.

所提出的研究总体上符合非交换几何的框架，特别是与鲍姆-康尼斯猜想、群分析和表示论相关的研究。鲍姆-康尼斯猜想将几何、拓扑和代数联系起来。从一个角度来看，它提出了一种理解群（一部分）表示空间的代数拓扑（K理论）的方法。虽然可以有效地描述（半简单）李群的所有表示，但对于一般的离散群来说，这项任务是不可能的。在这里，我们建议使用几何（非正曲率）和边界为大类离散组构建显式表示族。它们直接解决重要问题（沙洛姆猜想），涉及鲍姆-康尼斯猜想的现有方法以及离散群的调和分析。所提出的途径结合了解析群论和几何群论、李群表示论和随机游走的思想。该项目的理念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。我们建议为非正弯曲群的突出类别的（均匀有界）表示构造一个“紧凑图”。首先，我们处理可以进行“组合调和分析”的情况，即群正确作用于（有限维）CAT（0）立方体复合体的情况。其次，我们提取该构造的主要特征并用双曲群执行它，从而建立沙洛姆的猜想。