Research in Geometric Group Theory

在几何群论领域提出了三个方案。首先，Mladen Bestvina和我建议验证我们的一个猜想，这个猜想位于逻辑、几何和群论的界面上。一个副产品将是对马尔塞夫40年来提出的一个问题的答案，这个问题问自由基团的哪些子群是可以定义的。我们的技术灵感来自于Zlil Sela对Tarski问题的几何方法。第二个项目是与迈克尔·汉德尔的合作。为了解决自由群的外自同构群的共轭问题，我们在前人工作的基础上提出了进一步的改进。最后，对于第三个项目，Bestvina和我建议执行一个方案，证明自由群的外自同构群不包含任何高阶格。几何群论是一个令人兴奋的相对年轻的数学领域。一般的想法是从另一个数学领域(通常是代数)中提取一个问题，将问题转化为几何术语，然后使用几何方法来解决它。这一过程的一个说明性例子是我们如何教小学生加法。这组数字(一个代数对象)被解释为一条线上的一组点(一个几何对象)，加上2(一个代数运算)被解释为向右移动两个单位(一个几何变换)。我们建议对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。对于每一个由此产生的几何问题，我们都有有希望的攻击计划。

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.