Separability and logic in geometric group theory

AbstractAward：DMS-0906276首席研究员：亨利威尔顿该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。首席研究员将从事几条研究线，其动机是单词双曲组理论中的三个深度开放问题。 (1)每个双曲群都是剩余有限的吗？如果是这样，那么事实上字双曲群满足更强的可分性。PI的目的是研究这些分离性的已知的例子字双曲群和相关的群体，包括3-manifoldgroups和相对双曲群。(2)无挠双曲群的初等理论是可判定的吗？与丹尼尔格罗夫斯一起，PI打算通过证明Sela结果的算法版本来给出肯定的答案。 (3)每个字双曲群都包含一个曲面子群吗？关于Gromov的这个著名问题，即使对于一些非常基本的字双曲群的例子，也知之甚少。PI提出使用各种新技术来寻找字双曲群的例子中的表面子群，包括沿着最大循环子群的自由群的加倍。群是对称性的集合，例如保持整数格的欧几里得平面的所有平移的集合。 这个组是由一个单位长度的翻译在北，南，东，或西方向产生的，在这个意义上，任何组的元素可以通过组成这四个基本翻译的副本。 每一个随机生成的群都带有一个元素对之间距离的概念，通过计算将群中的一个元素带到另一个元素所必须使用的生成器的数量来定义。 这些项目集中在字双曲群上，它具有非欧几何中双曲平面的许多性质，并且已知是如此普遍，以至于随机选择的双曲生成群几乎肯定是字双曲的。 该奖项由拓扑学和基金会项目共同资助。

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.