Research in geometric group theory

获奖：DMS 1406167，首席研究员：Mark E. feighg在几何群论领域提出了两个项目，这是一个相对年轻的数学分支领域，在这个领域中，来自其他数学领域的问题被用几何术语重新表述，然后（希望）使用几何学者的工具包来解决。这种方法已经成功地解决了许多不同领域的问题，如组合群体理论（想想像魔方这样的情况，允许一组离散的移动）和逻辑。这两个项目都集中在一个特定的经典兴趣群，称为“自由群的外部自同构群”，表示为Out(F)。群是物体对称的集合，而Out(F)是自由群F的对称集合，而自由群F是一个重要的群，所有其他的群都可以从这个群中构造出来。任何一个项目的完成都将是该领域的重大进展。第一个项目是与犹他大学的Mladen Bestvina合作，继续研究Out(F)的几何形状。特别地，我们有一个计划来证明Out(F)具有有限的渐近维数。这项工作的一部分也将与犹他大学的帕特里克·雷诺兹共同完成。Out(F)的几何是目前的一个热门话题，其兴趣很大程度上是由Bestvina-Feighn和Handel-Mosher的结果所激发的，即Out(F)作用的某些空间是双曲的。第二个项目，与雷曼学院的Michael Handel合作，是为了证明Out(F)有一个可解的共轭问题。共轭问题是马克思·德恩在1911年左右提出的一个著名的可决性问题，它可以问任何群。它仍然对Out(F)开放，这一事实揭示了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.