Polish group actions: Dichotomies and generalized elementary embeddings
Polish group actions: Dichotomies and generalized elementary embeddings
复制标题
波兰群行动：二分法和广义基本嵌入
DOI：
--
复制
发表时间：
1998
期刊：
影响因子：
--
通讯作者：
H. Becker
中科院分区：
文献类型：
--
作者：
H. Becker
本文的结果涉及波兰群体行为描述理论中的两个不同主题。 Becker-Kechris [6] 一书是对该理论的介绍。我们的两个主题以及两个定理集合相当无关，但两个主题的证明本质上是相同的。局部紧波兰群，即第二可数局部紧群，是“拓扑群”领域的传统研究对象。最近，人们对非局部紧波兰群产生了兴趣，例如 S∞，N 上的对称群，拓扑为 NN 的子空间。本文的大部分内容涉及波兰群类的真子类，即那些承认完全左不变度量的波兰群。我们将这些称为 cli 组。 cli 组的类别包括所有局部紧凑组（命题 3.C.2 (d)），但还包括更多。例如，所有可解群都是 cli （Hjorth-Solecki；参见命题 3.C.2 (f)），并且 S∞ 的闭子群 G 是 cli，当且仅当 G 在 NN 中是封闭的（参见例 3.C.3 (a)）。我们研究波兰群在波兰空间上的连续行动以及相关的轨道等价关系。对于局部紧凑群体，我们所有的结果都是已知的，或者是微不足道的，或者两者兼而有之。但我们的一些结果可以被视为关于局部紧群的已知定理到更大类别的 cli 群的推广。本文考虑的两个主题之一是等价关系的二分定理。这些定理断言商空间要么“小”，要么包含特定“大”集合的副本。我们证明了 cli 群作用的轨道空间的一些二分定理。二分法有两种类型，我们称之为 Silver-Vaught 二分法和 Glimm-Effros 二分法。 Silver-Vaught 二分法断言，要么只有可数多个等价类，要么存在一组完美的成对不等价元素。假设连续统假设是否定，Silver-Vaught 二分法等价于存在可数多个等价类或 2א0 个等价类的命题。开放的拓扑沃特猜想是：对于波兰群 G 在波兰空间 X 上的任何连续作用，轨道等价关系满足 Silver-Vaught 二分法。这个猜想暗示了模型理论中的沃特猜想，并且是由它引发的。的确，
The results in this paper involve two different topics in the descriptive theory of Polish group actions. The book Becker-Kechris [6] is an introduction to that theory. Our two topics—and two collections of theorems—are rather unrelated, but the proofs for both topics are essentially the same. Locally compact Polish groups, i.e., second countable locally compact groups, are the traditional objects of study in the field known as “topological groups”. More recently, there has been interest in nonlocally compact Polish groups, such as S∞, the symmetric group on N, topologized as a subspace of NN. Much of this paper is concerned with a proper subclass of the class of Polish groups, namely those Polish groups which admit a complete left-invariant metric. We call these cli groups. The class of cli groups includes all locally compact groups (Proposition 3.C. 2 (d)), but also much more. For example, all solvable groups are cli (Hjorth-Solecki; see Proposition 3.C.2 (f)), and a closed subgroup G of S∞ is cli iff G is closed in NN (see Example 3.C.3 (a)). We study continuous actions by Polish groups on Polish spaces and the associated orbit equivalence relation. For locally compact groups, all of our results are known, or trivial, or both. But some of our results can be viewed as generalizations of known theorems about locally compact groups to the larger class of cli groups. One of the two topics considered in this paper is dichotomy theorems for equivalence relations. These are theorems which assert that either the quotient space is “small” or else it contains a copy of a specific “large” set. We prove some dichotomy theorems for the orbit space of actions by cli groups. There are two types of dichotomies, which we call the Silver-Vaught Dichotomy and the Glimm-Effros Dichotomy. The Silver-Vaught Dichotomy asserts that either there are only countably many equivalence classes or else there is a perfect set of pairwise inequivalent elements. Assuming the negation of the continuum hypothesis, the Silver-Vaught Dichotomy is equivalent to the proposition that there are either countably many equivalence classes or 2א0 equivalence classes. The Topological Vaught Conjecture, which is open, is: for any continuous action by a Polish group G on a Polish space X , the orbit equivalence relation satisfies the Silver-Vaught Dichotomy. This conjecture implies—and was motivated by—the Vaught Conjecture in model theory. Indeed,