Geometric Group Theory

摘要：在目前的项目中，首席研究员将继续研究各种有趣的类群可以作用的拓扑结构。特别是，PI将展示如何使各种类型的Artin群、Coxeter群、braid群、burnside群和纯对称自同构群作用于构造为具有一个或多个附加性质的空间。构造这样的空间的能力一直是几何群理论家的目标。由于这些群中的许多都是作为配置空间的基本群出现的，因此所提出的工作的基本模式可以总结如下：一个拓扑定义的空间，其基本群是感兴趣的，它作用于一个欧几里得细胞复合体，其结构被理解得足够好，可以对所涉及的群产生计算结果。这个项目的目的是增加这种类型的程序可以应用的群体的数量和种类。这种类型的项目延续了最近在几何群理论中对数学的实用和计算方面的强调。所考虑的结果将不仅表明各种对象的存在，而且将提供它们的构造算法和所涉及过程的复杂性界限。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得几何群论与理论计算机科学以及这些技术在现实世界中的应用密切相关。

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.