Asymptotic Methods in Geometric Group Theory

现代群论是一个发展非常迅速的数学领域，它使用了代数、几何、组合学、拓扑学、概率论、逻辑学和计算机科学的方法。从物理和化学到网络安全，群论的应用在整个科学中无处不在。该奖项支持群论中渐近方法领域的合作研究，自格罗莫夫关于双曲群和群的渐近不变量的开创性论文以来，该领域一直非常活跃。该项目将通过解决几个悬而未决的问题，极大地促进对该地区的了解。研究人员将为研究生提供建议，为本科生组织研究经验，并组织关于群论中渐近方法主题的会议。本研究项目中研究的问题包括：*估计亚标尔群的Dehn函数；*构造类似塔斯基怪物群的双曲群的作用；*构造允许高度传递作用的简单群；*描述群的所有可能的塔尔斯基数；以及*找到一个所有渐近锥都是树形的服从群。该项目的结果将促进对群论中渐近方法的理解。

Modern group theory is a very fast developing area of mathematics that uses methods from algebra, geometry, combinatorics, topology, probability, logic, and computer science. Applications of group theory are ubiquitous throughout science, from physics and chemistry to cyber security. This award supports collaborative research in the area of asymptotic methods in group theory, an area that has been extremely active since seminal papers by Gromov on hyperbolic groups and asymptotic invariants of groups. This project will significantly advance knowledge of the area by solving several open problems. The investigators will advise graduate students, organize research experiences for undergraduate students, and organize conferences on the subject of asymptotic methods in group theory.The problems under study in this research project include* estimating Dehn functions of metabelian groups;* constructing actions of hyperbolic groups resembling Tarski monster groups;* constructing simple groups that admit highly transitive actions;* describing all possible Tarski numbers of groups; and* finding an amenable group with all asymptotic cones tree-graded.The results of the project will advance understanding of asymptotic methods in group theory.