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).

几何群体理论领域提出了三个相当雄心勃勃的项目。第一个是由tarski逻辑问题的Zlil Sela的几何解的启发。 PI和Mladen Bestvina计划完成众多计划解决的计划意味着Mal'Cev的两个40年历史的问题。他们还建议将其方法概括为Tarski问题。第二个项目是迈克尔·汉德尔（Michael Handel），是为了解决自由群体F的外态形态群（F）的结合问题。这一领域的许多研究都由线性群体，surfacemapping class组之间的相似性和OUT（F）驱动。其他两个类别的共轭问题已经解决了。最终项目再次与Mladen Bestvina有关，是更好地了解自由群体外态构成群体的几何形状。特别是，提议在映射类集合设置中表现出类似于曲线复合物的复杂性是单词双曲线。在几何组理论领域提出了三个项目。几何群体理论是一个相对年轻的电学分支，其中数学其他领域的问题浮出水面，然后以几何术语解决。这种方法在有时被视为远处的区域中成功。例如，Zlil Sela使用几何方法来解决Tarski的肛门逻辑问题。受Sela方法的启发，Pi和Mladen Bestvina建议解决Mal'Cev的两个旧逻辑问题。群体理论是这些方法已取得了特定成功的另一个领域。小组在数学和这些方面无处不在。分子的对称性集就是一个例子。 Isan重要的组一类称为“自由组”，可以从中构建所有其他组。自由群体的重要组的对称集合（f），这是当前利益的主题。 PI提出了其他两个集中于Onof（F）的项目。在一个中，他和迈克尔·汉德尔（Michael Handel）提议解决一个旧的和基础问题（F），即它的共轭问题可以解决。在另一个项目中，PI和BESTVINA建议更好地理解OUT（F）的内在几何形状。