Geometric Group Theory

提议：DMS-9971682PI：乔纳森·麦卡蒙摘要：在本项目中，首席研究员将继续研究各种感兴趣的基团可以作用于其上的拓扑结构。特别是，PI将展示各种类型的Artin群、Coxeter群、辫子群、Burnside群和纯对称自同构群如何作用于被构造为具有一个或多个附加性质的空间。构造这种空间的能力长期以来一直是几何群论家的目标。由于这些群中的许多群是作为配置空间的基本群而产生的，因此许多拟议工作的模式可以概括如下：使其基本群感兴趣的拓扑学定义的空间作用于近似欧几里德胞格复形，其结构被足够好地理解以获得所涉及的群的计算余序列。这个项目的目的是增加可以应用这类程序的群的数量和种类。这类项目延续了几何群论中最近对数学的实际的、计算的方面的重视。正在考虑的结果不仅将显示各种对象的存在，而且将为它们的构造和所涉及的程序的复杂性的界限提供算法。因此，所描述的所有工作都可以作为执行这些类型计算的计算机软件包来实现。这种对算法及其复杂性的明确强调使几何群论接近于理论计算机科学和这些技术的现实应用。

Proposal: DMS-9971682PI: Jonathan McCammondAbstract: In the current project, the Principal Investigator will continue hisstudy of the topological structures on which various interestingclasses of groups can act. In particular, the PI will show howvarious types of Artin groups, Coxeter groups, braid groups, Burnsidegroups, and pure symmetric automorphism groups can be made to act onspaces constructed to have one or more additional properties. Theability to construct such spaces has long been a goal of geometricgroup theorists. Since many of these groups arise as the fundamentalgroups of configuration spaces the pattern underlying much of theproposed work can be summarized as follows: a topologically definedspace whose fundamental group is of interest is made to act on apiecewise Euclidean cell complex whose structure is understood wellenough to have computational comsequences for the group involved. Theaim of this project is to increase the number and variety of groups towhich thiss type of procedure can be applied.This type of project continues the recent emphasis within geometricgroup theory on the practical, computational aspects of mathematics.The results under consideration will not merely show the existence ofvarious objects, but will provide algorithms for their constructionand bounds of the complexity of the procedures involved. Thus all ofthe work described can be implemented as a computer software packagewhich performs these types of computations. This explicit emphasis onthe algorithms and their complexity makes geometric group theory aclose relative of theoretical computer science and of the real worldapplications of these techniques.