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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759115
• 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.

这是一项主要针对数理逻辑的提案，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，该计划集中于几个核心主题。近年来的学科，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和自动连续现象。这部分将集中于发现结构性的流动。 1990 年代 Gardner 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就是 G。 -equidecomposable，那么它们是使用可测量片段的 G-equidecomposable。这是对超有限等价关系的最新突破性结果的深远概括。超有限等价关系出现在遍历理论和描述性集合论的界面上，作为由整数 Z 群的 Borel 作用导出的等价关系，该学科中一个臭名昭著的开放问题询问是否有一个几乎处处超有限的等价关系。概率测度必须是超有限的。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）动作是可测群理论的中心主题之一。遍历维数是根据保留动作的概率测度的结构定义的群的不变量，可树化（强可树化）群是那些遍历（强遍历）维数等于 1 的群。臭名昭著的开放问题之一是询问是否每个可树化群。是强可树化的。自同构和有限拓扑的可拓性质 在 1990 年代，Hrushovski 证明了关于有限部分自同构的可拓的基本定理。图：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 的所有部分自同构都扩展到 G' 的自同构。从那时起，了解哪些 Fraisse 类就成为了扩展研究的焦点。有限结构具有这一性质。该领域最有趣的问题之一是 Herwig 和 Lascar 提出的一个长期问题：有限锦标赛类是否具有这一性质。自动连续性现象是 的性质。一个拓扑群，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 理论的联系已经证明了许多无限维群的同态群的自动连续性。