Mathematical Sciences: Geometry and Low-Dimensional Topology in Group Theory

数学科学：群论中的几何和低维拓扑

• 批准号: 9703756
• 负责人: Robert Friedman
• 金额: $ 10.5万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-01 至 2000-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9703756&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometry; Low; Dimensional

9703756 Sela Zlil Sela intends to complete his sequence of papers on the isomorphism problem for hyperbolic groups and the sequence of papers on automorphisms of free groups. He further plans to complete writing his work on the structure of sets of solutions to systems of equations in a free group, and on families of such systems depending on a set of defining parameters. This work is based on borrowing notions and ideas from algebraic geometry and low dimensional topology and implementing them in group theory. Sela and E. Rips (Hebrew University) are engaged in a long-standing investigation of fundamental algebraic structures known as ``groups,'' structures which abstract and capture the essence of symmetry. One of three key problem of the field is the isomorphism problem, the problem of telling when two groups, presented differently, are really the same. It is known that there can be no general recipe for solving this problem, one that will work for a completely arbitrary pair of groups. For this reason, recipes have been sought that work within large classes of interesting groups, and Sela is closing in on such a recipe for the class of (Gromov) hyperbolic groups, groups whose interest derives from being attached to geometric objects known as manifolds, both arbitrary manifolds of dimension three and negatively curved ones of any dimension. The ideas and techniques evolved in this work have led to other questions and to unexpected answers, all bearing on the deep and intimate connection between the algebra of groups and the geometry of manifolds. During the last year, Sela has developed a geometric approach to sets of solutions of equations in a so-called free group. The approach borrows heavily from algebraic geometry. More recently he has pushed this geometric approach further in order to study indexed families of sets of solutions of equations in groups. Sela and Rips now expect that the structure theory they have developed will find many applications in group theory and in mathematical logic. In particular, they believe that the tools they have in hand will provide positive answers to several questions posed by the logician Alfred Tarski around 1950. Since their techniques borrow heavily from low dimensional topology, they hope that their structure theory will also find applications to basic questions in low-d topology. ***

Zlil Sela打算完成他关于双曲群同构问题的一系列论文和关于自由群自同构的一系列论文。他进一步计划完成他关于自由群中方程组的解集结构的工作，以及依赖于一组定义参数的方程组族的工作。这项工作是基于借鉴代数几何和低维拓扑的概念和思想，并在群论中实现它们。Sela和E. Rips（希伯来大学）长期致力于研究被称为“群”的基本代数结构，这些结构抽象并捕捉了对称的本质。该领域的三个关键问题之一是同构问题，即判断两个不同呈现的群体是否真的相同的问题。众所周知，没有解决这个问题的通用方法，它将适用于完全任意的一对组。由于这个原因，人们一直在寻找在大量有趣群中工作的方法，Sela正在接近（Gromov）双曲群类的这种方法，这些群的兴趣来自于与被称为流形的几何物体相关联，无论是三维的任意流形还是任何维度的负弯曲的流形。在这项工作中发展的思想和技术导致了其他问题和意想不到的答案，所有这些都与群的代数和流形的几何之间的深刻而密切的联系有关。去年，Sela开发了一种几何方法来解决所谓的自由群中的方程组的解。这种方法大量借用了代数几何。最近，他进一步推动了这种几何方法，以研究群中方程解集的索引族。Sela和Rips现在期望他们发展的结构理论将在群论和数理逻辑中找到许多应用。特别是，他们相信，他们手中的工具将为逻辑学家阿尔弗雷德·塔斯基（Alfred Tarski）在1950年左右提出的几个问题提供积极的答案。由于他们的技术大量借鉴了低维拓扑，他们希望他们的结构理论也能应用于低维拓扑中的基本问题。＊＊＊