Boundary representations of non-positively curved groups

所提出的研究总体上符合非对易几何的框架，特别是与Baum-Connes猜想、群分析和表示理论相关的研究。鲍姆-康尼斯猜想将几何学、拓扑学和代数联系起来。从一个角度出发，提出了一种理解群表示空间(一部分)的代数拓扑(K-理论)的方法。虽然有效地描述(半单)李群的所有表示是可能的，但这项任务通常对于离散群是不可能的。在这里，我们建议使用几何(非正曲率)和边界来构造大类离散群的显式表示族。它们直接解决了重要的问题(沙洛姆猜想)，涉及到鲍姆-康尼斯猜想的现有方法，以及离散群上的调和分析。提出的途径结合了解析和几何群论、李群表示理论和随机游动的思想。本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。首先，我们讨论了可以进行“组合调和分析”的情形，即群作用在(有限维)CAT(0)立方体复数上的情形；其次，我们提取了这种构造的主要特征，并用双曲群来表示它，从而建立了Shalom猜想.

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.