Asymptotic Methods in Geometric Group Theory

几何群论中的渐近方法

• 批准号: 1500180
• 负责人: Alexander Olshanskiy
• 金额: $ 43.2万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-06-01 至 2019-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500180&HistoricalAwards=false
• 关键词: Asymptotic; Methods; Geometric; Group; Theory

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.

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、逻辑和计算机科学的方法。群论的应用在整个科学中无处不在，从物理和化学到网络安全。该奖项支持在群论中的渐近方法领域的合作研究，该领域自Gromov关于双曲群和群的渐近不变量的开创性论文以来一直非常活跃。该项目将通过解决几个开放的问题，大大提高该地区的知识。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的会议。本研究项目中研究的问题包括：* 估计亚阿贝尔群的Dehn函数;* 构造类似于Tarski怪物群的双曲群的作用;* 构造允许高度传递作用的简单群;* 构造允许高度传递作用的双曲群。* 描述所有可能的塔斯基数的群体;和 * 找到一个顺从的群体与所有渐近锥树分次。该项目的结果将推进理解的渐近方法在群论。