Research in geometric group theory

几何群论研究

• 批准号: 1406167
• 负责人: Mark Feighn
• 金额: $ 18.69万
• 依托单位: Rutgers University Newark
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406167&HistoricalAwards=false
• 关键词: Research; geometric; group; theory

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.

Abstractaward：DMS 1406167，首席研究员：Mark E. Feighntwo项目是在几何组理论领域提出的，几何组理论是一个相对年轻的数学子场，其中其他数学领域的问题以几何学术语重新汇总，然后（希望）使用地理计的工具工具（希望）解决。这种方法已经成功地解决了组合群体理论等不同领域的问题（考虑到允许使用离散的动作的鲁比克立方体）和逻辑。这两个项目都集中在一个特定的经典兴趣上，称为“自由群体的外部自动形态群体”，该群体表示为（f）。群体作为对象的对称性集成，而OUT（F）是Free组F的对称性集合，这是一个重要的组，可以从中构造所有其他组。这两个项目的完成都将代表该领域的重大进步。与犹他大学的Mladen Bestvina联合的第一个项目是继续研究Out（F）的几何形状。特别是，我们计划表明（F）具有有限的渐近维度。这项工作的一部分也将与犹他大学的帕特里克·雷诺兹（Patrick Reynolds）联合。 OUT（F）的几何形状目前是一个热门话题，在很大程度上是由Bestvina-Feighn和Handel-Mosher的结果引起的兴趣，而某些空间（F）行为是双曲线的。与雷曼学院（Lehman College）的迈克尔·汉德尔（Michael Handel）联合的第二个项目是证明（f）有一个可解决的共轭问题。共轭问题是Max Dehn在1911年左右提出的一个著名的可决定性问题，可以询问任何群体。 （f）仍然开放的事实揭示了我们对该群体的基本理解的差距。