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: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=740200
• 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.

这是一个建议主要是在数理逻辑，涉及一些最活跃的领域，如描述集理论和模型理论的齐次结构。该提案延续了PI的研究计划。该计划集中在近年来在该学科中一直处于中心地位的几个主题，如Borel组合学，可测群论，Hrushovski属性及其与profinite群理论和自动连续现象的联系。第一部分.在顺从的群体中流动。这一部分将着重于发现测度空间上顺从群作用中的结构现象。Gardner在1990年代提出的一个著名的猜想是：如果G是一个顺从群，它以保测度的方式作用在一个概率测度空间上，那么只要这个空间的两个可测子集是G-等可分解的，那么它们也是G-等可分解的。这是对最近突破性成果的一个意义深远的概括。第二部分.超有限等价关系的结构。超有限等价关系作为整数群Z的Borel作用诱导的等价关系出现在遍历理论和描述集合论的界面上。一个臭名昭著的公开问题在这一学科问是否等价关系，几乎处处超有限的关于每一个概率测度必须是超有限的。树型等价关系的结构。保概率测度作用的结构是测度群理论的核心问题之一。遍历维数和强遍历维数是根据其概率测度保持作用的结构定义的群的不变量。可树（强可树）群是遍历（强遍历）维数等于1的群。一个臭名昭著的开放性问题是，是否每个可树组都是强可树的。 第四部分.自同构与profinite拓扑的扩张性质。在1990年代，Hrushovski证明了一个关于有限图的部分自同构扩张的基本定理：对于每个有限图G，存在一个包含G作为导出子图的有限图G'，使得G的所有部分自同构扩张为G'的自同构。从那时起，它一直是扩展研究的焦点，以了解有限结构的Fraisse类共享此属性。在这方面最有趣的问题之一是一个长期存在的问题Herwig和Lascar问类有限锦标赛是否有这个属性。第五部分自动连续性现象。自动连续性是一个拓扑群的性质，它说任何从这个群到一个可分群的同态都是连续的。它最近已经证明了许多无限维群通过连接到Fraisee理论。本计画的目的是发展证明同胚群自动连续性的技巧。