Geometric Group Theory via Geometric Combinatorics

通过几何组合的几何群论

• 批准号: 0405783
• 负责人: Jon McCammond
• 金额: $ 11.53万
• 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: 2004
• 资助国家: 美国
• 起止时间: 2004-07-15 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405783&HistoricalAwards=false
• 关键词: Geometric; Group; Theory; via; Combinatorics

The goal of this project is to bring various classes of groupscommonly studied by geometric group theorists (such as Artin groups,one-relator groups, small cancellation groups, word-hyperbolic groups,fundmental groups of ample twisted face pairing 3-manifolds, etc.)within range of one of the standard general theories (such aspiecewise Euclidean spaces of nonpositive curvature, conformalnonpositive curvature, or Garside structures) by constructingappropriate complexes on which they act. For example, there areemerging strategies for constructing high-dimensional nonpositivelycurved cube complexes for metric small cancellation groups as well asfor fundamental groups of ample twisted face pairing 3-manifolds, forconstructing 2-dimensional conformally CAT(0) complexes forone-relator groups, and for constructing Garside-like structures forarbitrary Artin groups. Geometric and enumerative combinatorics playa prominent role in the constructions themselves as well as in theestablishment of their major properties. In each case, the complexesconstructed and the approaches themselves are new and innovative andtheir key properties are in the process of being established. Earlyindications are that these complexes carry geometric and combinatorialstructures which would resolve such longstanding conjectures as thecoherence of one-relator groups, and the solution of the word problemfor Artin groups.This project lies at the interface between geometric/combinatorialgroup theory and geometric/enumerative combinatorics. The formerstudies algebraic structures associated with geometric objects (suchas their group of symmetries) while the latter can be roughly definedas the study of things which can be described using only a finiteamount of data. This is precisely the type of mathematics thatcomputers can do. Computational mathematics might seem far removedfrom geometric considerations, but there is a growing collection ofcombinatorial phenomena which can best be viewed as finite analoguesof facts about the curvature of smooth spaces. The primary goal ofthis project is to use these computationally discovered combinatorialphenomena to construct complexes which carry a geometric/topologicalstructure which then explain the observed algebraic behavior of theoriginal groups.

本计画的目标是透过建构适当的复合体，将几何群论家经常研究的各类群（如Artin群、单相关群、小对消群、字双曲群、充足的扭曲面配对3流形的基本群等）纳入标准一般理论（如非正曲率的欧几里得空间、共形非正曲率或Garside结构）的范围内。例如，对于度量小抵消群和大量扭曲面配对3流形的基本群，有构造高维非正弯曲立方体配合物的新策略；对于单相关群，有构造二维共形CAT(0)配合物的新策略；对于任意Artin群，有构造Garside-like结构的新策略。几何和计数组合学在构造本身以及确定其主要性质方面起着突出的作用。在每一个案例中，所构建的综合体和方法本身都是新的和创新的，它们的关键属性正在建立过程中。早期的迹象表明，这些复合体具有几何和组合结构，这将解决一些长期存在的猜想，如单亲缘群的相干性，以及解决类群的单词问题。本项目处于几何/组合群论和几何/枚举组合学的交叉点。前者研究与几何对象相关的代数结构（例如它们的对称组），而后者可以大致定义为研究只能使用有限数量的数据来描述的事物。这正是计算机所能做的数学运算。计算数学似乎与几何考虑相距甚远，但有越来越多的组合现象可以被最好地视为光滑空间曲率事实的有限类比。该项目的主要目标是利用这些计算发现的组合现象来构建具有几何/拓扑结构的复合体，然后解释观察到的原始群的代数行为。