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

这个项目的目标是理解Thompson群的度量性质。汤普森定义了一个令人着迷的群族，它出现在广泛的数学学科中，包括逻辑、同伦理论和离散群的测度论。汤普森的群往往是许多奇怪的群论现象中最简单或唯一已知的例子。尽管许多研究人员进行了大量研究，但人们对这些群体的许多特性知之甚少。这个项目通过更好地理解Thompson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基础上，他们在一些情况下合作，在另一些情况下单独工作，与汤普森群的各种度量性质有关。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·克利里和奥尔巴尼大学的詹妮弗·塔巴克访问巴塞罗那加泰罗尼亚理工大学的何塞·布里洛提供旅行支持。这项数学研究属于群论领域。群是一种抽象的结构，通常可以将其可视化为几何对象的对称性的集合。一组这样的对称的一些共同性质可以用系统的方法抽象出来。在本研究中，群本身被作为几何对象来研究，几何洞察力有助于更有效地理解某些群的代数结构。这项研究的重点是一个被称为汤普森群体的家族，以第一个定义它们的研究人员的名字命名。汤普森的群可以用二叉树对来几何地理解，这是一种将这个群与理论计算机科学中的问题联系起来的方法。这个项目的更广泛的影响包括这些应用于计算机科学中与二叉树的形状有关的算法问题，国际参与和对类似项目感兴趣的研究人员之间的交流，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.