US-SPAIN Cooperative Research: Metric Properties of Thompson's Group

这个项目的目标是理解汤普森群的度量属性。汤普森定义了一个引人入胜的群族，它出现在广泛的数学学科中，包括逻辑学、同伦理论和离散群的度量理论。汤普森群通常是一些奇怪的群论现象的最简单或唯一已知的例子。尽管许多研究人员进行了大量的研究，但对这些群体的许多特性知之甚少。本项目通过更好地理解这些群的Cayley图来研究Thompson群的度量结构。这项研究建立在研究人员早期的结果的基础上，这些研究人员在某些情况下共同工作，在其他情况下单独工作，涉及汤普森群的各种度量属性。该奖项支持美国和西班牙在数学方面的合作研究。它为纽约城市学院的肖恩·克利里（Sean Cleary）和奥尔巴尼大学的詹妮弗·塔巴克（Jennifer Taback）访问巴塞罗那加泰罗尼亚理工大学的何塞·布里洛（Jose Burillo）提供旅行支持。这项数学研究属于群论领域。群是一种抽象结构，通常可以将其可视化为几何对象的对称集合。一组这样的对称性的一些共同性质可以用系统的方式抽象出来。在本研究中，群本身是作为几何对象来研究的，几何洞察力可以帮助我们更有效地理解一些群的代数结构。这项研究的重点是一个被称为汤普森群体的群体，以第一个定义它们的研究者的名字命名。汤普森群可以用二叉树对从几何上理解，这种方法将这个群与理论计算机科学中的问题联系起来。该项目的广泛影响包括计算机科学中与二叉树形状相关的算法问题的应用，对类似项目感兴趣的研究人员之间的国际参与和交流，以及可能的学生参与研究。

The goal of this project is the understanding of the metric properties of Thompson's groups. Thompson defined a fascinating family of groups, which appear in a wide range of mathematical disciplines, including logic, homotopy theory and the measure theory of discrete groups. Thompson's groups are often the simplest or only known examples of a number of strange group-theoretic phenomena. Despite a great deal of study by a number of researchers, many properties of these groups are poorly understood. This project investigates the metric structure of Thompson's groups by developing a better understanding of the Cayley graphs of these groups. This research builds upon earlier results of the researchers, working together in some cases and separately in others, relating to various metric properties of Thompson's groups.This award supports US-Spanish cooperative research in mathematics. It provides travel support for Sean Cleary of The City College of New York and Jennifer Taback of the University at Albany to visit Jose Burillo of the Universitat Politecnia de Catalunya in Barcelona. This mathematical research lies in the area of group theory. A group is an abstract structure, which can often be visualized as a collection of symmetries of a geometrical object. Some common properties of a set of such symmetries can be made abstract in systematic ways. In this research, groups themselves are studied as geometric objects, and geometric insight can help to understand the algebraic structure of some groups more effectively. This research focuses on a family of groups known as Thompson's groups, after the researcher who first defined them. Thompson's groups can be understood geometrically using pairs of binary trees, an approach that relates this group to questions in theoretical computer science. The broader impacts of this project include these applications to algorithmic questions in computer science related to the shape of binary trees, international involvement and exchange between researchers interested in similar projects, and possible student involvement in research.