HARMONICITY AND RIGIDITY OF DISCRETE AND ARITHMETIC GROUPS

提出的活动的主要目的是加强对离散群的刚性理论的研究，特别强调重要的一类算术群。本研究的大部分内容旨在通过一个具体的一般猜想，将算术群理论中的各种结果精化、重铸和统一为一个新的设置，特别强调了30年前的马古利斯-齐默猜想（关于公约子群的结构）作为一个特例。提出的工作中的一个关键概念是调和性，它是与该猜想相关的中心成分，以及与群上的Lipschitz调和函数相关的刚性的不同建议方向（由PI最近与Terry Tao的联合工作中使用它们所激发的）。所谓的算术群理论是现代数学中最深刻、最美丽、最富有成果和被长期研究的理论之一。它涉及纯数学的大多数领域，从表示、数和k理论的代数方面，到各种形式的几何，再到遍历理论和动力学的分析方面。建议的工作旨在解决一些基本问题，并建议对部分理论进行重铸，从而提供新的亮点并促进该领域的新发展。特别强调的是格雷戈里·马古利斯和罗伯特·齐默30年前的猜想。它的解决方案预计将涉及新的动力学思想，这与谐性（粗略地说，一种平均下稳定的状态）密切相关。新的研究和应用的和谐性有望带来进展的其他方向提出的工作。

The principal aim of the proposed activity is to enhance research in the rigidity theory of discrete groups, with particular emphasis on the important class of arithmetic groups. Substantial parts of the research aim to sharpen, recast and unify diverse results in the theory of arithmetic groups into one novel setting, through a concrete general conjecture, with particular emphasis on the 30-year old Margulis-Zimmer conjecture (concerning the structure of commensurated subgroups), as a special case. A key notion in the proposed work is that of harmonicity, which emerges as central ingredient in relation to thisconjecture, as well as in a different suggested direction of rigidity pertaining to Lipschitz harmonic functions on groups (motivated by their use in recent joint work of the PI with Terry Tao).The theory of so-called arithmetic groups is one of the deepest, most beautiful, fruitful, and long studied ones in modern mathematics. It involves most fields of pure mathematics, from the algebraic side of representation, number, and K-theory, through geometry in its various forms, to the analytic side of ergodic theory and dynamics. The proposed work aims to address some fundamental issues and suggest a recast of parts of the theory, thereby shedding new light and enhancing new developments in the field. Of particular emphasis is a 30-year old conjecture due to Gregory Margulis and Robert Zimmer. Its resolution is expected to involve new dynamical ideas, which are strongly related to harmonicity (roughly speaking, a state which is stable under averaging). Novel investigations and applications of harmonicity are expected to bring progress in other directions of the proposed work.