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

9703756 Sela Zlil Sela打算完成他关于双曲群的同构问题的论文序列和关于自由群自同构的论文序列。他还计划完成他关于自由群中的方程组的解集的结构的工作，以及关于依赖于一组定义参数的这种系统的族的工作。这项工作的基础是借鉴代数几何和低维拓扑的概念和思想，并在群论中加以实现。Sela和E.Rips(希伯来大学)长期致力于基本代数结构的研究，这种结构被称为“群”，它抽象并捕捉到对称的本质。该领域的三个关键问题之一是同构问题，即当两个呈现不同的组实际上相同时的判断问题。众所周知，不可能有解决这个问题的通用配方，一个完全适用于任意一对群体的配方。出于这个原因，人们一直在寻找在大类有趣的群中工作的配方，而Sela正在接近这样一类(Gromov)双曲群的配方，这种群的兴趣源于对称为流形的几何对象的依附，既有三维的任意流形，也有任何维的负曲线流形。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，所有这些都与群代数和流形几何之间的深刻而密切的联系有关。在过去的一年里，SELA开发了一种几何方法来求解所谓的自由群中的方程组。这种方法大量借鉴了代数几何。最近，为了研究群组方程解集的指标族，他进一步推进了这种几何方法。塞拉和里普斯现在预计，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·塔尔斯基在1950年左右提出的几个问题提供肯定的答案。由于他们的技术大量借鉴了低维拓扑学，他们希望他们的结构理论也能应用于低维拓扑学中的基本问题。***

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. ***