Research in Geometric Group Theory

几何群论研究

• 批准号: 0805440
• 负责人: Mark Feighn
• 金额: $ 16.26万
• 依托单位: Rutgers University Newark
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0805440&HistoricalAwards=false
• 关键词: Research; Geometric; Group; Theory

Three projects are proposed in the area of geometric group theory. In the first, Mladen Bestvina and I propose to verify a conjecture of ours that lies at the interface of logic, geometry, and group theory. A by-product would be an answer to a forty-year-old question of Mal'cev who asks which subgroups of free groups are definable. Our techniques are inspired by Zlil Sela's geometric approach to the Tarski problem. The second project is with Michael Handel. We propose to build on our previous work in order to solve the conjugacy problem for outer automorphism groups of free groups. Finally, for the third project Bestvina and I propose to carry out a scheme we have for showing that outer automorphism groups of free groups don't contain any higher rank lattices.Geometric group theory is an exciting and relatively young area of mathematics. The general idea is to take a problem from an another area of mathematics (most often algebra), translate the problem into geometric terms, and then use the methods of geometry to solve it. An illustrative example of this process is how we teach addition to elementary school students. The set of numbers (an algebraic object) is interpreted as the set of points on a line (a geometric object) and addition by say two (an algebraic operation) is interpreted as a shift to the right by two units (a geometric transformation). We propose to carry out such a translation process for three well-known problems, one coming from logic and two from algebra. We have promising plans of attack for each of the resulting geometric problems.

几何群体理论领域提出了三个项目。首先，我和Mladen Bestvina提议验证我们的猜想，该猜想位于逻辑，几何和群体理论的界面。副产品将是对Mal'Cev的四十岁问题的答案，该问题询问哪些自由团体可以定义。我们的技术灵感来自Zlil Sela对Tarski问题的几何方法。第二个项目是迈克尔·汉德尔（Michael Handel）。我们建议以我们以前的工作为基础，以解决外在自由构成群体的共轭问题。最后，对于第三个项目，Bestvina和我提议执行我们拥有的计划，以表明外部自动组的自由组不包含任何更高的等级晶格。几何群体理论是一个令人兴奋且相对年轻的数学领域。一般的想法是从数学的另一个领域（最常见的代数）处理问题，将问题转化为几何术语，然后使用几何方法来解决它。此过程的一个说明性示例是我们如何教给小学生的补充。一组数字（代数对象）被解释为一条线上的一组点（几何对象），而用两个单位则将两个单位（几何转换）解释为两个（代数操作）的添加（代数操作）。我们建议为三个众所周知的问题进行这样的翻译过程，一个来自逻辑，两个来自代数。我们为每个产生的几何问题制定了有希望的攻击计划。