Research in Geometric Group Theory

在几何群论领域提出了三个项目。在第一篇文章中，我和Mladen Bestvina提议验证我们的一个猜想，这个猜想位于逻辑、几何和群论的交界处。一个副产品是回答40年前Mal'cev提出的一个问题，即自由群的哪些子群是可定义的。我们的技术灵感来自Zlil Sela解决塔斯基问题的几何方法。第二个项目是和Michael Handel合作的。为了解决自由群的外自同构群的共轭问题，我们提出在前人工作的基础上。最后，对于第三个项目，Bestvina和我提出了一个方案来证明自由群的外自同构群不包含任何高秩格。几何群论是一个令人兴奋的和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）中获取问题，将问题转化为几何术语，然后使用几何方法来解决它。这个过程的一个说明性的例子是我们如何教小学生加法。一组数字（一个代数对象）被解释为一条直线上的点的集合（一个几何对象），加上两个（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。对于每一个几何问题，我们都有很好的解决方案。

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.