Research in geometric group theory

几何群论研究

• 批准号: 1105193
• 负责人: Mark Feighn
• 金额: $ 27.59万
• 依托单位: Rutgers University Newark
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1105193&HistoricalAwards=false
• 关键词: Research; geometric; group; theory

Three rather ambitious projects are proposed in the area of geometricgroup theory. The first is inspired by Zlil Sela's geometric solutionof Tarski's logic problem. The PI and Mladen Bestvina plan to completetheir program to solve by geometric means two forty-year-old problemsof Mal'cev. They also propose to generalize their methods and providean alternate proof to the Tarski problem. The second project, jointwith Michael Handel, is to solve the conjugacy problem for the outerautomorphism group Out(F) of a free group F. Much of the research inthis area is driven by the similarities between linear groups, surfacemapping class groups, and Out(F). The conjugacy problem has beensolved for these other two classes. The final project, again jointwith Mladen Bestvina, is to better understand the geometry of outerautomorphism groups of free groups. In particular, it is proposed to showthat a complex analogous to the curve complex in the mapping classgroup setting is word hyperbolic.Three projects are proposed in the area of geometric grouptheory. Geometric group theory is a relatively young branch ofmathematics in which problems from other areas of mathematics arereformulated and then solved in geometric terms. This approach hasbeen successful in areas that are sometimes viewed as distant fromgeometry. For example, Zlil Sela used geometric methods to solve anold logic problem of Tarski. Inspired by Sela's methods, the PI andMladen Bestvina propose to solve two old logic problems ofMal'cev. Group theory is another area where these methods have beenparticularly successful. Groups are ubiquitous in math and thesciences. The set of symmetries of a molecule is an example. There isan important class of groups called "free groups" from which all othergroups can be constructed. The set of symmetries of a free group F isanother important group, denoted Out(F), which has been the subject ofmuch current interest. The PI proposes two other projects focusing onOut(F). In one, he and Michael Handel propose to solve an old andfundamental problem on Out(F), namely that its conjugacy problem issolvable. In the other project, the PI and Bestvina propose to betterunderstand the intrinsic geometry of Out(F).

在几何群论领域提出了三个相当雄心勃勃的计划。第一个是受Zlil Sela对Tarski逻辑问题的几何解法的启发。PI和Mladen Bestelf计划完成他们的计划，通过几何手段解决两个40岁的Malcev问题。他们还提出推广他们的方法，并提供替代证明塔斯基问题。第二个项目是与Michael Handel合作，解决自由群F的外自同构群Out（F）的共轭问题。这一领域的许多研究都是由线性组、表面映射类组和Out（F）之间的相似性驱动的。另外两类的共轭性问题已得到解决。最后一个项目再次与Mladen Bestvina联合，是为了更好地理解自由群的超自同构群的几何形状。特别地，本文提出了在映射类群的条件下，与曲线复形类似的复形是双曲的，并在几何群论领域提出了三个方案。几何群论是一个相对年轻的数学分支，其中的问题，从其他领域的数学重新制定，然后解决几何方面。这种方法在有时被视为远离几何学的领域取得了成功。例如，Zlil Sela使用几何方法解决了Tarski的一个旧逻辑问题。受Sela方法的启发，PI和Mladen Bestelf提出解决Mal 'cev的两个老逻辑问题。群论是这些方法特别成功的另一个领域。在数学和科学中，群体无处不在。分子的对称性集合就是一个例子。有一类重要的群叫做“自由群”，所有其他群都可以从它构造出来。自由群F的对称集是另一个重要的群，记为Out（F），它是当前人们非常感兴趣的课题。PI提出了另外两个项目，重点是Out（F）。在第一篇文章中，他和Michael Handel提出解决Out（F）上的一个古老而基本的问题，即它的共轭问题是可解的。在另一个项目中，PI和Besthart建议更好地理解Out（F）的内在几何结构。