Geometric Group Theory and the Topology of 3-Manifolds

DMS-0203883G。彼得·斯科特几何群论与3-流形的拓扑学提出者计划继续与Gadde Swarup合作研究几何群论与3-流形的拓扑学之间的联系。上世纪70年代，S完成了紧致三维流形的特征子流形理论。从上世纪80年代中期的S开始，人们开始清楚地看到，在纯代数环境下也有类似的拓扑学结果。在过去的15年里，几位作者对此进行了研究，但到目前为止，除了非常特殊的情况外，没有一个结果与拓扑学的结果真正相似。现在，提出者和Swarup已经产生了拓扑学结果的第一个真正的代数模拟，他们计划继续他们的研究，以看看他们的结果可以进行到什么程度。他们的论点和结果已经为3-流形的拓扑学提供了新的见解，并导致了几何群论的新结果。作者期望在这一领域的进一步工作将对几何群论产生重大影响，并计划继续与Gadde Swarup合作研究群论与三维流形理论之间的联系。大约40年前，斯图林斯的一些非凡的、完全出人意料的工作首次展示了这些联系的深度。最近，几位作者的工作延续了这一主题，表明群可以按类似于三维流形的分解方式进行分解。提出者和斯瓦鲁普为这一领域提供了一种新的方法，他们计划继续他们的调查，看看他们的结果可以走多远。他们的论点和结果已经为三维流形理论提供了新的见解，并在群论中产生了新的结果。提出者期望这一领域的进一步工作将对群论产生重大影响。

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.