Research in Geometric Group Theory

几何群论研究

• 批准号: 0405623
• 负责人: Ruth Charney
• 金额: --
• 依托单位: Brandeis University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-15 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405623&HistoricalAwards=false
• 关键词: Research; Geometric; Group; Theory

This project involves problems in geometric group theory and spaces of non-positive curvature with a focus on Artin groups and Garside groups. Artin groups span a wide range of groups including braid groups, free groups, and free abelian groups. The first part of the project concerns automorphism groups of right-angled Artin groups. These are finitely generated groups whose presentations involve only commutator relations between generators. By varying the number of commutator relations, right-angled Artin groups may be viewed as "interpolating" between free groups and free abelian groups. The automorphism groups of free groups have been the focus of much research in recent years. They have been shown to have much in common with linear groups, in particular with the automorphism groups of free abelian groups. This project aims to put these results in a broader context by studying the automorphism group of a general right-angled Artin group. The key idea is to construct a contractible space, analogous to Culler and Vogtmann's "outer space", on which the automorphism group acts.The second part of the project concerns Garside groups. Garside groups are groups with algorithmic properties similar to those of braid groups. A Garside structure on a group gives a very powerful tool for studying the combinatorial and geometric properties of the group. This approach has been used extensively in the study of finite type Artin groups. While no such structure exists for infinite type Artin groups, it appears that something close to a Garside structure may exist for at least some of these groups. The project will consider various generalizations of Garside groups and their properties.Symmetries of geometric objects have been studied since ancient times and have played an important role in many areas of mathematics and the sciences. They form the model for the abstract mathematical notion of a "group". Groups arise in nearly every field of mathematics. While not all groups occur naturally as symmetry groups, it is always possible to constructgeometric objects on which a given group acts as symmetries. In recent years, this interplay between groups and geometry has been increasingly exploited to understand various types of infinite groups. A particularly interesting class of groups that lends itself to such techniques are the "braid groups". The n-strand braid group, as the name suggests, encodes the different ways that a set of n strings can be braided. It also describes the different ways that a collection of n particles can move around in a plane. Braid groups have applications to topology, mathematical physics, and cryptography. Another fundamental class of groups are the "free groups". These groups form the foundation for studying algorithmic and combinatorial properties of groups in general. In this project we will study a large class of groups, known as Artin groups, that includes the braid groups, the free groups, and many others. The project aims to reach a better understanding of Artin groups by studying geometric objects associated with them, as well as searching for new algorithmic techniques.

该项目涉及几何群论和非正曲率空间问题，重点关注 Artin 群和 Garside 群。 Artin 群涵盖广泛的群，包括辫子群、自由群和自由阿贝尔群。该项目的第一部分涉及直角 Artin 群的自同构群。这些是有限生成群，其表示仅涉及生成器之间的换向器关系。通过改变交换子关系的数量，直角Artin群可以被视为自由群和自由阿贝尔群之间的“插值”。自由群的自同构群近年来一直是许多研究的焦点。它们已被证明与线性群有很多共同点，特别是与自由交换群的自同构群。 该项目旨在通过研究一般直角 Artin 群的自同构群，将这些结果置于更广泛的背景下。关键思想是构造一个可收缩空间，类似于卡勒和沃格特曼的“外层空间”，自同构群在其上起作用。该项目的第二部分涉及加赛德群。 Garside 组是具有与辫子组类似的算法属性的组。群上的 Garside 结构为研究群的组合和几何性质提供了非常强大的工具。这种方法已广泛用于有限型 Artin 群的研究。虽然无限型 Artin 群不存在这样的结构，但至少其中一些群似乎可能存在接近 Garside 结构的结构。该项目将考虑加赛德群及其性质的各种推广。几何对象的对称性自古以来就被研究，并在数学和科学的许多领域发挥了重要作用。它们形成了“群”这一抽象数学概念的模型。数学组几乎出现在每个领域。虽然并非所有群都自然地作为对称群出现，但始终可以构造给定群充当对称性的几何对象。近年来，群与几何之间的相互作用越来越多地被用来理解各种类型的无限群。适合这种技术的一类特别有趣的组是“辫子组”。 n 股编织组，顾名思义，对一组 n 个字符串的不同编织方式进行编码。 它还描述了 n 个粒子的集合在平面上移动的不同方式。辫子群在拓扑学、数学物理和密码学中都有应用。 另一个基本的团体类别是“自由团体”。这些群构成了研究群的算法和组合属性的基础。在这个项目中，我们将研究一大类群，称为 Artin 群，其中包括辫子群、自由群和许多其他群。 该项目旨在通过研究与Artin群相关的几何对象以及寻找新的算法技术来更好地理解Artin群。