Growth, Gap, and Geometry

增长、差距和几何

• 批准号: 1611758
• 负责人: Jing Tao
• 金额: $ 15.4万
• 依托单位: University of Oklahoma Norman Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1611758&HistoricalAwards=false
• 关键词: Growth; Gap; Geometry

Award: DMS 1611758, Principal Investigator: Jing TaoGeometry is a basic area of human research, originating with our visual awareness of the world; it is present implicitly or explicitly in every scientific discipline. The discovery of symmetries and their usefulness is arguably one the greatest achievements of human knowledge. Concretely, one of the most effective ways of understanding and controlling geometric shapes is to exploit their symmetries. The modern approach is to encode this information in something we call the fundamental group. One goal of topologists and geometric group theorists is to understand all possible fundamental groups, classify them, and study their own intrinsic geometric properties. Classically, the most studied geometric objects have been in low dimensions, particularly in dimension two. This is called surface theory and is one of the focus areas of the proposed projects. The principal investigator is also interested in taking the intuitions that one gains from studying the fundamental groups of surfaces and extrapolating them to more complex objects. This area is called geometric group theory, an increasingly active subject within mathematics over the last few decades.The research projects address a broad spectrum of research problems in the general areas of geometric group theory and Teichmuller theory, with interactions with hyperbolic geometry, low-dimensional topology, and dynamics. The specific topics include: (1) Growth tightness of group actions on metric spaces. This notion was first introduced by Grigorchuk and de la Harpe for word metrics and it has connections to the Hopfian property for groups and the Rank Rigidity Conjecture. (2) Stable commutator lengths in right-angled Artin groups and related groups. (3) The geometry of Teichmuller space equipped with the Thurston metric.

奖项：DMS 1611758，首席研究员：陶晶几何是人类研究的基础领域，起源于我们对世界的视觉认知；它或隐或显地存在于每一个科学学科中。 对称性的发现及其用途可以说是人类知识最伟大的成就之一。具体来说，理解和控制几何形状的最有效方法之一是利用它们的对称性。现代方法是将这些信息编码为我们称为基本组的东西。拓扑学家和几何群理论学家的目标之一是理解所有可能的基本群，对它们进行分类，并研究它们本身的内在几何性质。传统上，研究最多的几何对象都是低维的，特别是二维的。 这称为表面理论，是拟议项目的重点领域之一。首席研究员还对从研究基本表面组中获得的直觉并将其外推到更复杂的物体感兴趣。这个领域被称为几何群论，是过去几十年数学中日益活跃的学科。这些研究项目解决了几何群论和泰希米勒理论一般领域的广泛研究问题，以及与双曲几何、低几何的相互作用。维度拓扑和动力学。具体主题包括：（1）度量空间上群行动的增长紧度。这个概念首先由 Grigorchuk 和 de la Harpe 引入用于词度量，它与群的 Hopfian 性质和Rank 刚性猜想有联系。 (2) 直角Artin群及相关群中的稳定换向器长度。 (3) 配备瑟斯顿度量的Teichmuller空间的几何结构。

项目成果

Genus bounds in right-angled Artin groups

直角 Artin 群中的属界

• DOI: 10.5565/publmat6412010
• 发表时间: 2020
• 期刊: Publicacions Matemàtiques
• 作者: Forester, Max;Soroko, Ignat;Tao, Jing
• 通讯作者: Tao, Jing

Effective quasimorphisms on right-angled Artin groups

直角 Artin 群的有效拟同构

• DOI: 10.5802/aif.3277
• 发表时间: 2019
• 期刊: Annales de l'Institut Fourier
• 作者: Fernós, Talia;Forester, Max;Tao, Jing
• 通讯作者: Tao, Jing