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等方法。从物理、化学到网络安全，群论在科学领域的应用无处不在。该奖项支持群论中渐近方法领域的合作研究，自Gromov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尚未解决的问题，显著推进对该领域的认识。研究人员将指导研究生，为本科生组织研究经验，并组织关于群论渐近方法的会议。本课题研究的问题包括：代谢群Dehn函数的估计；*构造类似塔斯基怪群的双曲群动作；*构建允许高度传递行为的简单群体；*描述群的所有可能的塔斯基数；和*求一个具有所有渐近锥的可服从群。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.