Discrete and continuous geometry in group theory

群论中的离散和连续几何

• 批准号: 0805716
• 负责人: Jon McCammond
• 金额: $ 22.97万
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0805716&HistoricalAwards=false
• 关键词: Discrete; continuous; geometry; group; theory

Artin groups are a large class of groups commonly studied by geometricgroup theorists. They are related to Coxeter groups, they include andgeneralize the braid groups, and they have a natural definitioninvolving complexified hyperplane complements. Despite theseaffiliations with well-known objects, very few Artin groups arecurrently understood, even at a basic level, and those that are belongto highly restrictive families defined primarily by the situationswhere existing techniques can be applied. The approach described hereis fundamentally new and different. The goal of this proposal is tounderstand the natural, continuous geometric objects (recentlyintroduced by the principal investigator and his collaborators) thathave the potential to serve as a geometric foundation for the study ofall Artin groups in a uniform fashion. These geometric objects,called factor geometries, are of interest in their own right. Thereis, for example, a natural continuous group G that has the orthogonalgroup as a quotient, contains the braid group as a subgroup, and isthe fundamental group of a finite-dimensional metric simplicialcomplex whose universal cover is the contractible factor geometry witha building-like structure on which G acts. This unusual uncountablegroup and space, defined via the collection of minimal factorizationsof spherical isometries into reflections, is a hybrid mix of Liegroups and metric simplicial complexes, and every Artin group acts ona similarly defined continuous geometric object. The eventualstructure theory should resemble that of Lie groups and earlyindications are that these types of complexes carry geometric andcombinatorial structures sufficient to resolve many longstandingconjectures about arbitrary Artin groups.Mathematical objects, like many physical objects, can be betterunderstood when we fully understand the symmetries they possess. Thealgebraic structure that records how these symmetries interact iscalled a ``group'' and the groups under consideration here are a classof groups generated by ``reflections'' (a symmetry like the reflectedimage one sees through a mirror). The main goal of this project is todeepen our understanding of the relationship between symmetry groupsbuilt from reflections (specifically Coxeter groups) and a secondclass of symmetry groups, called Artin groups. The most famousexample of an Artin group is the braid group, the group that keepstrack of the distinct ways in which several strands of string can bebraided together. Constructions involving Coxeter groups, Artingroups and braid groups proliferate throughout mathematics, includingsome recent mathematical physics. In many of these cases, theimmediate connections are, strictly speaking, to Artin groups ratherthan Coxeter groups, but these connections have not been pursuedpartly because the theory of Artin groups is underdeveloped. Oncethis situation is rectified within geometric group theory, the nextstep will be to export the resulting structure theory to theseneighboring domains

Artin群是几何群理论家经常研究的一大类群。 它们与Coxeter群有关，包含并推广了辫群，并且有一个涉及复化超平面补的自然定义。 尽管这些分支与著名的对象，很少阿廷集团是目前理解的，即使在一个基本的水平，和那些被称为高度限制性的家庭主要是由现有技术可以应用的情况下定义。 这里所描述的方法是全新的和不同的。 这个提议的目的是理解自然的、连续的几何对象（最近由主要研究者和他的合作者介绍），这些对象有可能作为以统一的方式研究所有阿廷群的几何基础。 这些几何对象称为因子几何，它们本身就很有趣。 例如，有一个自然连续群G，它有正交群作为商，包含辫子群作为子群，并且是有限维度量单纯复形的基本群，其通用覆盖是G作用于其上的具有类似建筑物结构的可收缩因子几何。 这个不寻常的不可数群和空间，通过将球面等距分解为反射的最小因式分解的集合来定义，是李群和度量单纯复形的混合体，并且每个Artin群都作用于类似定义的连续几何对象。 事件结构理论应该类似于李群理论，早期的迹象表明，这些类型的复形具有几何和组合结构，足以解决许多关于任意Artin群的长期问题。数学对象，像许多物理对象一样，当我们完全理解它们所具有的对称性时，可以更好地理解它们。 记录这些对称性如何相互作用的代数结构被称为“群”，这里所考虑的群是一类由“反射”（一种对称性，就像人们通过镜子看到的反射图像）产生的群。 这个项目的主要目标是加深我们对从反射（特别是Coxeter群）和第二类对称群（称为Artin群）建立的对称群之间关系的理解。 阿廷群最著名的例子是辫群，它跟踪几股绳子编织在一起的不同方式。 涉及Coxeter群，Artinggroups和braid群的构造在整个数学中激增，包括一些最近的数学物理。在许多情况下，直接的联系，严格地说，Artin群，而不是Coxeter群，但这些联系没有被追求，部分原因是Artin群的理论是欠发达的。 一旦这种情况在几何群论中得到纠正，下一步将是将所得到的结构理论导出到邻近的领域