Actions and invariants of residually finite groups

该建议解决了剩余有限群领域中的各种渐近问题。该方案的核心目标是具有平凡交的有限指标子群的降链。这些链通过自同构在局部有限根树上产生作用。在这种树的边界上的遍历作用和子群的陪集空间的有限作用之间存在着有趣的相互作用。人们可以为这些链和行为赋予各种自然不变量，这些链和行为最终被证明与群论中的众所周知的概念有关，如顺从性、Betti数、扩张器、有界生成、成本、大小或自相似。项目中使用的方法借鉴了拓扑学、概率论、数论、遍历理论和分析等工具。除了群论，解决所提出的问题还将对遍历理论、数论和拓扑学产生影响。任何结构的对称集合形成一个群。因此，每当一个人需要分析对称性时(例如，在化学或量子物理中，或者在数学中，在拓扑学、几何学、数论和码论中)，群论自然就会出现。拟议项目的主要目标是理解自然出现的数学对象，根据对它们所附基团的各种估计。除了在遥远的数学原理之间建立令人惊讶的联系，从而产生深刻而有用的结果外，该项目还具有基本的方面，可以在本科生水平上展示，从而服务于教育目的。建议的一部分是制定并开始一门新的高级代数探究式课程，从而导致实验教材的出版。

The proposal addresses various asymptotic questions in the realm of residually finite groups. The core object of the proposal is a descending chain of finite index subgroups with trivial intersection. These chains give rise to actions on locally finite rooted trees by automorphisms. There is an interesting interplay between the ergodic action on the boundary of such a tree and the finite actions of the coset spaces of the subgroups. One may assign various natural invariants to these chains and actions that turn out to relate to well-known notions in group theory, like amenability, Betti numbers, expanders, bounded generation, the cost, largeness or self-similarity. The methods used in the project borrow tools from topology, probability, number theory, ergodic theory and analysis. Besides group theory, solving the proposed problems would have impact on ergodic theory, number theory and topology. The set of symmetries of any structure forms a group. Thus group theory naturally comes into the picture whenever one needs to analyze symmetries (for instance in chemistry or quantum physics, or inside mathematics, in topology, geometry, number theory and the theory of codes). The main goal of the proposed project is to understand naturally appearing mathematical objects in terms of various estimates on the groups attached to them. Besides establishing surprising connections between distant mathematical principles and thus leading to deep and useful results, the project also has elementary aspects that can be presented on an undergraduate level and thus serve educational purposes. Part of the proposal is to work out and start a new high level Inquiry Based course in algebra, leading to the publication of experimental teaching material.