Geometric Group Theory and the Topology of 3-Manifolds

dms - 0203883 g。Peter scott几何群论和3流形的拓扑学提议者计划继续他与Gadde Swarup在几何群论和3流形拓扑学之间的联系方面的合作工作。在20世纪70年代，完成了紧凑三流形的特征子流形理论。从20世纪80年代中期开始，这种拓扑结果在纯代数环境中出现了类似的结果。在过去的15年里，有几位作者对此进行了研究，但到目前为止，除了在非常特殊的情况下，没有任何结果与拓扑结果真正相似。现在，提出者和swarupp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研究，以看看他们的结果可以进行多远。他们的论点和结果已经对3流形的拓扑结构提供了新的见解，并在几何群论中产生了新的结果。该领域的进一步研究将对几何群论产生重大影响。他计划继续与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，斯托林斯的一些非凡的、完全出乎意料的工作首次证明了这些联系的深度。最近，几位作者的工作继续了这一主题，表明群可以以一种与三维流形可以分解的方式密切相似的方式分解。提议者和Swarup在这一领域提出了一种新方法，他们计划继续调查，看看他们的结果能推广到什么程度。他们的论点和结果已经为三维流形理论提供了新的见解，并导致了群论的新结果。提案人期望在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.