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.

拟议的研究通常符合非交通性几何形状的框架，特别是与鲍姆（Baum）猜想，对组的分析和代表理论有关的研究。鲍姆 - 康涅狄格州的猜想连接几何，拓扑和代数。从一种角度来看，它提出了一种理解（一部分）代表空间的代数拓扑（K理论）的方法。虽然可以有效地描述（半imple）谎言组的所有表示，但总体而言，这项任务是不可能的。在这里，我们建议使用几何形状（非阳性曲率）和边界构建大量离散组的明确表示。他们直接解决了重要的问题（Shalom的猜想），与鲍姆（Baum）的现有方法有关 - 康涅狄格州的猜想以及对离散组的谐波分析。所提出的途径结合了分析和几何群体理论，谎言群体的表示理论和随机步行的想法。该项目的理念是利用并进一步发展几何组理论与分析/非合并几何形状之间的联系。 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的猜想。