Polish Group Actions: Dichotomies and Generalized Elementary Embeddings

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,