Geometric Group Theory and the Topology of 3-Manifolds

几何群论和3-流形拓扑

• 批准号: 0203883
• 负责人: G. Peter Scott
• 金额: $ 22.85万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-08-01 至 2006-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0203883&HistoricalAwards=false
• 关键词: Geometric; Group; Theory; Topology; Manifolds

DMS-0203883G. Peter ScottGeometric Group Theory and the Topology of 3-ManifoldsThe proposer plans to continue his joint work with Gadde Swarup onconnections between geometric group theory and the topology of3-manifolds. In the 1970's, the theory of the characteristic submanifold of a compact 3-manifold was completed. Starting in themid 1980's, it began to be apparent that there were analogues of this topological result in the purely algebraic setting. Several authors have worked on this in the last 15 years, but so far none of the results have been truly analogous to the topological results except in very special cases. Now the proposer and Swaruphave produced the first true algebraic analogues of the topological results and they plan to continue their investigationsin order to see how far their results can be carried. Their arguments and results have already given new insights into the topology of 3-manifolds and led to new results in geometric grouptheory. The proposer expects that further work in this area willhave a significant impact on geometric group theory.The proposer plans to continue his joint work with Gadde Swarup onconnections between group theory and the theory of 3-dimensionalmanifolds. About 40 years ago, some remarkable, and entirely unexpected, work of Stallings first demonstrated the depth of these connections. More recently, work of several authors continued this theme by showing that groups can be decomposed in afashion closely analogous to the way in which 3-dimensional manifolds can be decomposed. The proposer and Swarup have given a new approach to this area and they plan to continue their investigations in order to see how far their results can be carried. Their arguments and results have already given new insights into the theory of 3-dimensional manifolds and led to newresults in group theory. The proposer expects that further work inthis area will have a significant impact on group theory.

DMS-0203883G。彼得斯科特几何群论和拓扑的3流形的提议者计划继续他的联合工作与加德Swarup之间的连接几何群论和拓扑的3流形。在20世纪70年代，紧致3-流形的特征子流形理论得到了完善。从20世纪80年代中期开始，人们开始发现在纯代数环境中有类似的拓扑结果。几位作者在过去的15年中对此进行了研究，但到目前为止，除了在非常特殊的情况下，没有一个结果真正类似于拓扑结果。现在，提出者和Swarup已经产生了拓扑结果的第一个真正的代数类似物，他们计划继续他们的调查，以了解他们的结果可以进行多远。他们的论点和结果已经给了新的见解的拓扑3流形和导致新的结果几何群论。提议者预计该领域的进一步工作将对几何群论产生重大影响。提议者计划继续与加德·斯瓦鲁普（Gadde Swarup）共同研究群论与3维流形理论之间的联系。大约40年前，Stallings的一些引人注目的、完全出乎意料的工作首次证明了这些联系的深度。最近，工作的几个作者继续这一主题表明，集团可以分解的afashion密切类似的方式，其中3维流形可以分解。提议者和Swarup已经提出了一种新的方法来处理这一问题，他们计划继续进行调查，以了解他们的结果能走多远。他们的论点和结果已经为三维流形理论提供了新的见解，并导致了群论中的新结果。提议者期望在这一领域的进一步工作将对群论产生重大影响。