Boundary representations of non-positively curved groups

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、群分析和表征理论相关的研究。Baum- Connes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表示空间（一部分）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）和边界来构造大类别离散群的显式表示族。他们直接解决了重要的问题（Shalom猜想），涉及到Baum- Connes猜想的现有方法，以及离散群的谐波分析。提出的路径结合了解析和几何群论、李群表示理论和随机行走的思想。该项目的理念是利用并进一步发展几何群论与分析/非交换几何之间的联系。我们提出构造非正弯曲群的突出类的（一致有界）表示的“紧图”。首先，我们处理可以进行“组合调和分析”的情况，即群适当作用于（有限维）CAT(0)立方配合物的情况。其次，我们提炼了结构的主要特征，并用双曲群进行了构造，从而建立了沙洛姆猜想。

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.