Measurable group theory, descriptive set theory and model theory of homogeneous structures

可测群论、描述集合论和齐次结构模型论

• 批准号: RGPIN-2020-05445
• 负责人: Sabok, Marcin
• 金额: $ 2.26万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=716339
• 关键词: Measurable; group; theory; descriptive; set

This is a proposal primarily in mathematical logic, concerning some of its most active areas, such as descriptive set theory and model theory of homogeneous structures. The proposal continuous a research program of the PI. This program concentrates on several themes that have been central in the discipline in the recent years, such as Borel combinatorics, measurable group theory, the Hrushovski property with its connections to profinite group theory and the automatic continuity phenomenon. Part I. Flows in amenable groups. This part will focus on discovering structural phenomena in actions of amenable group on measure spaces. A well-known conjecture of Gardner from the 1990's states that if G is an amenable group acting on a probability measure space in a measure preserving way, then whenever two measurable subsets of the space are G-equidecomposable, then they are G-equidecomposable using measurable pieces. This is a far-reaching generalization of recent breakthrough results. Part II. Structure of hyperfinite equivalence relations. Hyperfinite equivalence relations appear on the interface of ergodic theory and descriptive set theory as equivalence relations induced by Borel actions of the group Z of the integers. A notorious open problem in this discipline asks whether an equivalence relation which is almost everywhere hyperfinite with respect to every probability measure must be hyperfinite. Part III. Structure of treeable equivalence relations. Structure of p.m.p (probability measure preserving) actions of finitely generated groups is one of the central themes in measured group theory. Ergodic dimension and strong ergodic dimension are invariants of a group defined in terms of the structure of its probability measure preserving actions. Treeable (strongly treeable) groups are those of ergodic (strong ergodic) dimension equal to 1. One of the notorious open questions asks if every treeable group is strongly treeable. Part IV. Extension properties for automorphisms and profinite topology. In the 1990's Hrushovski proved a fundamental theorem about extensions of partial automorphisms of finite graphs: for every finite graph G there exists a finite graph G' containing G as an induced subgraph such that all partial automorphisms of G extend to automorphisms of G'. Since then it has been a focus of extended study to understand which Fraisse classes of finite structures share this property. One of the most interesting problems in this area is a long-standing question of Herwig and Lascar asking whether the class of finite tournaments has this property. Part V. The automatic continuity phenomenon. Automatic continuity is the property of a topological group which says that any homomorphism from that group into a separable group is continuous. It has been recently proved for many infinite-dimensional groups via connections to Fraisee theory. This project aims at developing techniques for proving the automatic continuity for homeomorphism groups.

这主要是在数理逻辑中的一个提议，涉及其一些最活跃的领域，例如描述性集合论和齐次结构的模型论。该提案延续了 PI 的研究计划。该课程集中于近年来该学科的几个核心主题，例如波雷尔组合学、可测群论、赫鲁索夫斯基性质​​及其与有限群论的联系以及自动连续现象。 第一部分：顺从的群体流动。这部分将集中于发现服从群在测度空间上的行为中的结构现象。 Gardner 在 1990 年代提出的一个著名猜想指出，如果 G 是一个以测度保持方式作用于概率测度空间的服从群，那么只要该空间的两个可测子集是 G 等可分解的，那么它们就是 G 等可分解的，使用可测量的部分。这是对近期突破性成果的深远概括。 第二部分。超有限等价关系的结构。超有限等价关系出现在遍历理论和描述集合论的接口上，作为由整数组 Z 的 Borel 作用导出的等价关系。该学科中一个臭名昭著的开放性问题是，对于每个概率测度几乎处处超有限的等价关系是否一定是超有限的。 第三部分。可树化等价关系的结构。有限生成群的 p.m.p（概率测度保持）行为的结构是测度群理论的中心主题之一。遍历维数和强遍历维数是根据其概率测度保留动作的结构定义的群的不变量。可树化（强可树化）组是那些遍历（强遍历）维数等于 1 的组。臭名昭著的开放问题之一是询问是否每个可树化组都是强可树化的。 第四部分。自同构和有限拓扑的扩展性质。 1990年代，Hrushovski证明了关于有限图的部分自同构扩展的基本定理：对于每个有限图G，都存在一个包含G作为诱导子图的有限图G'，使得G的所有部分自同构扩展为G'的自同构。从那时起，了解哪些 Fraisse 类有限结构具有这一性质就成为了广泛研究的焦点。该领域最有趣的问题之一是 Herwig 和 Lascar 提出的一个长期存在的问题，即有限锦标赛类别是否具有此属性。 第五部分自动连续现象。自动连续性是拓扑群的属性，它表示从该群到可分离群的任何同态都是连续的。最近通过与 Fraisee 理论的联系，它已被证明适用于许多无限维群。该项目旨在开发证明同胚群自动连续性的技术。