Research in geometric group theory

摘要奖：DMS 1406167，主要研究者：Mark E.几何群论是一个相对年轻的数学分支，在这个分支中，其他数学领域的问题被重新用几何术语表述，然后（希望）使用几何工具箱解决。这种方法已经成功地解决了组合群论（想想魔方这样的情况，其中允许一组离散的移动）和逻辑等不同领域的问题。这两个项目都集中在一个特定的组的经典利益称为“外自同构群的自由群”这是表示出（F）。群作为对象的对称性集合而出现，Out（F）是自由群F的对称性集合，自由群F是一个重要的群，所有其他群都可以从它构造出来。第一个项目是与犹他州大学的Mladen Bestelf合作，继续研究Out（F）的几何。特别地，我们计划证明Out（F）具有有限的渐近维数。这项工作的一部分也将联合帕特里克雷诺兹在犹他州大学。Out（F）的几何是目前的一个热门话题，人们的兴趣在很大程度上是由Bestvina-Feighn和Handel-Mosher的结果引起的，他们认为Out（F）作用于某些空间是双曲的。第二个项目是与莱曼学院的迈克尔·亨德尔合作，证明Out（F）有一个可解的共轭问题。共轭问题是一个著名的可判定性问题，由马克斯·德恩在1911年左右提出，可以对任何群提出。事实上，它仍然为Out（F）开放，这表明我们对这个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.