Research in geometric group theory

奖项：DMS 1406167，首席研究员：马克·E·费恩在几何群论领域提出了两个项目，几何群论是一个相对年轻的数学子领域，数学中其他领域的问题被重新表述为几何术语，然后(希望)使用几何学家的工具包来解决。这种方法已经成功地解决了组合群论(想一想像魔方这样的情况，其中允许一组离散的动作)和逻辑等不同领域的问题。这两个项目都集中在一个特殊的经典兴趣群上，这个群被称为“自由群的外自同构群”，它被表示为(F)。群作为对象的对称集而产生，Out(F)是自由群F的对称集，它是一个重要的群，所有其他的群都可以从它构造出来。这两个项目的完成都代表着该领域的重大进步。第一个项目是与犹他大学的Mladen Bestvina合作，继续研究Out(F)的几何。特别地，我们有一个计划来证明OUT(F)有有限的渐近维度。这项工作的一部分还将与犹他大学的帕特里克·雷诺兹合作。Out(F)的几何目前是一个热门话题，引起人们兴趣的很大程度上是由于Bestvina-Feighn和Handel-Mosher的结果，即Out(F)作用于某些空间是双曲的。第二个项目是与雷曼学院的迈克尔·汉德尔合作，旨在证明OUT(F)存在一个可解的共轭问题。共轭问题是马克斯·德恩在1911年左右提出的一个著名的可决定性问题，可以问到任何群体。它仍然对OUT(F)开放的事实表明，我们对这一群体的基本理解存在差距。

AbstractAward: DMS 1406167, Principal Investigator: Mark E. FeighnTwo projects are proposed in the area of geometric group theory, a relatively young subfield of mathematics where problems from other areas of math are reformulated in geometric terms and then (hopefully) solved using a geometer's toolkit. This approach has been successful in solving problems in such diverse areas as combinatorial group theory (think of a situation such as Rubik's cube where a discrete set of moves is allowed) and logic. Both projects are focused on a particular group of classical interest called "the outer automorphism group of a free group" which is denoted Out(F). Groups arise as sets of symmetries of objects and Out(F) is the set of symmetries of a free group F, an important group from which all others can be constructed. Completion of either project would represent a major advance in the field.The first project, joint with Mladen Bestvina at the University of Utah, is to continue the study of the geometry of Out(F). In particular, we have a plan to show that Out(F) has finite asymptotic dimension. Part of this work will also be joint with Patrick Reynolds at the University of Utah. The geometry of Out(F) is currently a hot topic with interest spurred in large part by results of Bestvina-Feighn and Handel-Mosher that certain spaces on which Out(F) acts are hyperbolic. The second project, joint with Michael Handel at Lehman College, is to show that Out(F) has a solvable conjugacy problem. The conjugacy problem is a famous decidability question formulated by Max Dehn around 1911 that can be asked about any group. The fact that it remains open for Out(F) reveals a gap in our basic understanding of this group.