Research in Geometric Group Theory

几何群论研究

• 批准号: 0504917
• 负责人: Mark Feighn
• 金额: --
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0504917&HistoricalAwards=false
• 关键词: Research; Geometric; Group; Theory

An exciting recent development in geometric group theory is ZlilSela's beautiful work on the Tarski problem. This problem asks whetherthe elementary theory of a non-abelian free group is independent ofthe free group in question. Feighn proposes two projects in thisdirection, both joint with Mladen Bestvina. The first is acontinuation of a project to simplify Sela's work. In the second,representing a new direction, a solution to a modified form ofTarski's problem is proposed. This will lead to new information aboutdefinable sets (objects of primary importance) and will answerquestions of Rips and Sela. A third proposed project is to explore theextent to which the JSJ-decomposition of a group can be foundalgorithmically. The specific class of groups to be examined consistsof graphs of free groups, which are of current interest on a number offronts. With Guo-An Diao, the principal investigator has shown thatthe Grushko decomposition (a coarser decomposition than the JSJ) ofthese groups can be found algorithmically. There are two possiblesatisfactory resolutions, namely a construction of an algorithm toproduce the JSJ-decomposition of a group in the class or a proof thatthis problem is unsolvable. For this class of groups, which exists onthe boundary of what can be understood algorithmically, eitherresolution would be important.Geometric group theory is a relatively young branch of mathematicsthat uses geometric methods to understand groups. The idea to take aproblem in group theory, translate it into a problem in geometry, andthen use geometric methods to solve the translated problem. Theprincipal investigator proposes to use geometric methods to exploretwo general areas. The first is motivated by Zlil Sela's geometricsolution to the Tarski problem which on the face of it is a problem inthe intersection of group theory and logic. Roughly, the problem is todistinguish groups using the truth of statements formed using only theusual symbols of logic ("there exists", "for all", variables, etc) andgroup operations. For example, you can tell if a group is abelian byasking whether it is true that, for all elements x and y, xy=yx. Thefirst proposed project, joint with Mladen Bestvina, is to simplify andextend Sela's ideas. This project is ongoing and there has alreadybeen significant progress. The second proposed project follows anothertrend in group theory that is encapsulated by the question: How muchgroup theory can a computer understand? Much as molecules can bedecomposed into simpler pieces (atoms), groups can be often bedecomposed into simpler groups. The project is to understand suchdecompositions for a class of groups called "graphs of free groups", aclass which has attracted much recent attention. It turns out thatdecomposing groups translates into the geometric process of cutting aspace into simpler pieces. The goal is to use this geometric idea todiscover an algorithm that, given a group, finds its decomposition.The principal investigator, with Guo-An Diao, has already constructedan algorithm for a related process that can readily be implemented ona computer.

几何群体理论的最新发展是Zlilsela在Tarski问题上的精美作品。这个问题询问非亚伯自由群体的基本理论是否独立于相关的自由群体。 Feighn提出了这两个方向的两个项目，均与Mladen Bestvina联合。首先是对简化Sela作品的项目的实力。在第二个代表新方向的情况下，提出了一种解决改良形式的问题的解决方案。这将导致有关可定义集（主要重要性的对象）的新信息，并将回答撕裂和Sela的问题。第三个提议的项目是探索可以从基础上进行JSJ分解的扩展。要检查的特定组组组成的自由组的图表，这些图是当前在数量外的关注的组。对于郭安·迪亚奥（Guo-An Diao），主要研究人员表明，这些组的grushko分解（比JSJ更粗体）可以通过算法找到。有两种可能令人满意的分辨率，即算法构造了班级中一个组的JSJ分解，或者证明了这一问题是无法解决的。对于这类群体，存在于可以从算法上理解的边界上存在的，分析组理论是数学的相对年轻分支，这是使用几何方法来理解组的一个相对年轻的分支。在群体理论中采用问题的想法，将其转化为几何问题，然后使用几何方法来解决翻译问题。原理研究者建议使用几何方法来探索一般领域。第一个是由兹利尔·塞拉（Zlil Sela）对塔斯基（Tarski）问题的几何法的动机，从表面上讲，这是群体理论与逻辑的交集中的一个问题。粗略地，问题是使用仅使用逻辑的tausaual符号形成的语句的真实（“存在”，“所有”，“变量等”和“组”操作。例如，您可以判断一个组是否是Abelian，因为对于所有元素x和y，xy = yx是否真的。 与Mladen Bestvina联合的第一个项目是简化和扩展Sela的想法。该项目正在进行中，并且已经取得了重大进展。第二个提出的项目遵循小组理论中的另一个趋势，该问题由以下问题封装：计算机可以理解多少群体理论？正如分子可以将其固定成更简单的碎片（原子）一样，通常可以将基团组合成更简单的组。该项目是要了解一类称为“自由小组图”的小组的混合，aclass引起了最近的关注。事实证明，编译组转化为将asspace切成更简单的零件的几何过程。目的是使用这种几何思想来发现一种算法，鉴于一个小组，该算法发现了其分解。