Treeings, quasi-invariance, and ergodic combinatorics

树、拟不变性和遍历组合学

• 批准号: RGPIN-2020-07120
• 负责人: Tserunyan, Anush
• 金额: $ 2.11万
• 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=741608
• 关键词: Treeings; quasi; invariance; ergodic; combinatorics

GENERAL SCOPE The proposal lies within the broad scope of definable equivalence relations on Polish spaces, which is a modern focus of descriptive set theory. This theory provides a general framework for understanding the nature of classification of mathematical objects up to some notion of equivalence, and, due to its broad scope, it has natural interactions with many areas of mathematics. A central place in this theory is occupied by countable Borel equivalence relations (CBERs), which arise via actions of countable groups as well as via locally countable graphs. These connections between equivalence relations, group actions, and graphs create an extremely fruitful interplay between descriptive set theory, ergodic theory, measured group theory, and descriptive graph combinatorics. The overarching goal of the proposal is to deepen the understanding of these connections and further the theory of CBERs having major open questions as guiding targets. The projects of the proposal can be grouped into three topics. TREEINGS AND SUBTREEINGS In the study of CBERs, a principal role is played by those that are treeable, i.e. there is an acyclic Borel graph (a treeing) whose connected components are exactly the equivalence classes. These not only form a particularly interesting subclass of CBERs, but also serve as a critical tool for studying all CBERs in general. One project concerns a closure property of the class of treeable CBERs, while others require the development of new constructions of subtreeings, subgraphings, and subforests. The latter concerns the endeavor of structurally witnessing the nonamenability of a group within its free action on a probability space. To attack the main problems here, I propose the development of percolation theory on measured graphs. This is interesting on its own and opens new prospects and questions. QUASI-INVARIANCE A CBER $E$ is better understood when it is probability measure preserving (pmp), Borel automorphism of $E$ preserve a probability measure. This is because a number of techniques from various areas is available for pmp CBERs, including the theory of cost, $\ell^2$-(co)homology, operator algebras, and percolation theory. However, when the CBER is merely quasi-pmp, that is, every Borel automorphism of $E$ only preserves the non-nullness of sets, none of these techniques are available. In my recent work, I develop new tools for dealing with quasi-invariance, using which, I propose to generalize several results known in the pmp setting to the quasi-pmp setting. Furthermore, I have an idea for a definition of cost in this setting, which should be investigated. If promising, I propose developing a theory of quasi-pmp cost. ERGODIC THEOREMS The last topic is devoted to the pointwise ergodic theorems, namely, proving new instances and finding new proofs of known ones using a pointwise-combinatorial tiling argument in the style of my recent proof the Birkhoff ergodic theorem.

一般范围该建议在于波兰空间上的可定义等价关系的广泛范围内，这是描述集合论的现代焦点。这一理论提供了一个一般框架，用于理解数学对象的分类性质，直到某种等价概念，并且由于其广泛的范围，它与许多数学领域有着自然的相互作用。在这个理论中的一个中心位置被可数博雷尔等价关系（CBER）占据，它通过可数群的作用以及通过局部可数图产生。等价关系、群作用和图之间的这些联系在描述性集合论、遍历理论、度量群理论和描述性图组合学之间创造了极其富有成效的相互作用。该提案的总体目标是加深对这些联系的理解，并进一步发展以重大开放问题为指导目标的CBER理论。该提案的项目可分为三个专题。 树和子树在CBER的研究中，主要的角色是由那些可树的，即有一个非循环的Borel图（树），其连接组件正好是等价类。这些不仅形成了CBER的一个特别有趣的子类，而且也是研究所有CBER的关键工具。一个项目涉及一类可树CBERs的封闭属性，而另一些项目则需要开发新的子树、子图和子林结构。后者涉及的奋进，结构性地见证了一个群体在其概率空间上的自由行动的不顺从性。为了解决这里的主要问题，我提出了发展渗流理论的测量图。这本身就很有趣，并开辟了新的前景和问题。当CBER $E$是概率测度保持（pmp）时，它的拟不变性更好理解，$E$的Borel自同构保持概率测度。这是因为有许多不同领域的技术可用于pmp CBER，包括成本理论，$\ell^2 $-（上）同调，算子代数和渗透理论。然而，当CBER仅仅是准pmp时，也就是说，$E$的每个Borel自同构只保持集合的非空性，这些技术都不可用。在我最近的工作中，我开发了新的工具来处理准不变性，使用它，我建议推广几个已知的结果在pmp设置的准pmp设置。此外，我有一个关于在这种情况下成本定义的想法，应该加以研究。如果有希望的话，我建议发展准pmp成本理论。各态历经定理最后一个主题致力于逐点各态历经定理，即以我最近证明伯克霍夫各态历经定理的风格，使用逐点组合平铺论证来证明新实例并找到已知实例的新证明。