Separability and logic in geometric group theory

几何群论中的可分离性和逻辑

• 批准号: 0906276
• 负责人: Cameron Gordon
• 金额: $ 9.63万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0906276&HistoricalAwards=false
• 关键词: Separability; logic; geometric; group; theory

AbstractAward: DMS-0906276Principal Investigator: Henry WiltonThis award is funded under the American Recovery and ReinvestmentAct of 2009 (Public Law 111-5).The principal investigator will pursue several lines of researchmotivated by three deep open questions in the theory ofword-hyperbolic groups. (1) Is every hyperbolic group residuallyfinite? If so, then in fact word-hyperbolic groups satisfy muchstronger separability properties. The PI intends to investigatethese separability properties for known examples ofword-hyperbolic groups and related groups, including 3-manifoldgroups and relatively hyperbolic groups. (2) Is the elementarytheory of a torsion-free hyperbolic group decidable? Togetherwith Daniel Groves, the PI intends to give an affirmative answerby proving algorithmic versions of the results of Sela. (3) Doesevery word-hyperbolic group contain a surface subgroup? Little isknown about this famous question of Gromov, even for some verybasic examples of word-hyperbolic groups. The PI proposes to usevarious new techniques to find surface subgroups in examples ofword-hyperbolic groups, including doubles of free groups alongmaximal cyclic subgroups.A group is a collection of symmetries, for example the collectionof all translations of the Euclidean plane that preserve theinteger lattice. This group is generated by the translations ofone unit of length in the north, south, east, or west directions,in the sense that any element of the group can be obtained bycomposing copies of those four basic translations. Everyfinitely generated group carries a notion of distance betweenpairs of its elements, defined by counting the number ofgenerators that must be applied to carry one element of the groupto another. These projects concentrate on word-hyperbolicgroups, which have many of the properties of the hyperbolic planefrom non-Euclidean geometry and are known to be so common that arandomly selected finitely generated group is almost surelyword-hyperbolic. This award is jointly funded by the programs inTopology and Foundations.

奖项：dms -0906276首席研究员：Henry wilton该奖项由2009年美国复苏与再投资法案（公法111-5）资助。在双曲群理论中有三个深层次的开放问题，主要研究者将会进行几条研究路线。(1)是否每个双曲群都是残有限的？如果是这样，那么事实上词双曲群满足更强的可分性。PI打算研究已知的双曲群和相关群的这些可分离性，包括3-流形群和相对双曲群。(2)无扭双曲群的初等理论是可判定的吗？与丹尼尔·格罗夫斯（Daniel Groves）一起，PI打算通过证明Sela结果的算法版本来给出肯定的答案。(3)每个词双曲群是否都包含一个表面子群？关于格罗莫夫这个著名的问题，我们所知甚少，甚至对于一些非常基本的词双曲群的例子也是如此。本文提出了利用各种新技术在双曲群的例子中寻找曲面子群，包括自由群沿极大循环子群的双元。群是对称的集合，例如欧几里得平面的所有平移的集合，这些平移保持了整格。这个组是由北、南、东或西方向上的一个单位长度的翻译产生的，也就是说，这个组的任何元素都可以通过组合这四个基本翻译的副本来获得。每个有限生成的群都带有其元素对之间距离的概念，通过计算必须应用的生成器的数量来定义，以将群中的一个元素携带到另一个元素。这些项目集中在词双曲群上，它具有非欧几里得几何中双曲平面的许多特性，并且被认为是如此普遍，以至于随机选择的有限生成群几乎肯定是词双曲的。该奖项由拓扑学和基金会项目共同资助。