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.

最近几何群论的一个令人兴奋的发展是ZlilSela在Tarski问题上的出色工作。这个问题问的是非阿贝尔自由群的基本理论是否独立于所讨论的自由群。fehn在这个方向上提出了两个项目，都与Mladen Bestvina合作。首先是一个简化Sela工作的项目的延续。第二部分，提出了一种改进形式的tarski问题的解法，代表了一个新的方向。这将导致关于可定义集（最重要的对象）的新信息，并将回答Rips和Sela的问题。第三个提议的项目是探索组的jsj分解可以在多大程度上基于算法。要研究的群体的特定类别包括自由群体的图，它们在许多方面都是当前感兴趣的。与王国安一起，首席研究员已经证明，这些群体的Grushko分解（比JSJ更粗糙的分解）可以通过算法找到。有两种可能的令人满意的解决方案，即构造一种算法来生成类中一个组的jsj分解，或者证明这个问题是不可解的。对于这类存在于算法可以理解的边界上的群体，任何一种解决方案都很重要。几何群论是一个相对年轻的数学分支，它使用几何方法来理解群。把群论中的一个问题，转化为几何中的一个问题，然后用几何的方法来解决转化后的问题。首席研究员建议使用几何方法来探索两个一般领域。第一个是由Zlil Sela对Tarski问题的几何解所激发的，这个问题表面上是一个群论和逻辑交叉的问题。粗略地说，问题是使用仅使用通常的逻辑符号（“存在”、“对于所有”、变量等）和群运算形成的陈述的真值来区分群。例如，您可以通过询问对于所有元素x和y， xy=yx是否为真来判断一个组是否为阿贝尔组。第一个提议的项目是与Mladen Bestvina合作，旨在简化和扩展Sela的想法。这个项目正在进行中，已经取得了重大进展。第二个提议的项目遵循了群论的另一个趋势，这个趋势被一个问题所概括：计算机能理解多少群论？就像分子可以分解成更简单的分子（原子）一样，基团通常也可以分解成更简单的基团。这个项目是为了理解一类被称为“自由群图”的群的分解，这类最近引起了很多关注。事实证明，将群体分解转化为将空间切割成更简单块的几何过程。我们的目标是使用这个几何思想来发现一个算法，给定一个群，找到它的分解。首席研究员和刁国安已经为一个相关过程构建了一个算法，可以很容易地在计算机上实现。