Vaught's Conjecture and the Glimm-Effros property for Polish transformation groups

• Main Authors: Greg Hjorth, Slawomir Solecki
• Other Authors: The Pennsylvania State University CiteSeerX Archives
• Format: Text
• Language: English
• Published: 1995
• Subjects: sami
• Online Access: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.191.383; http://www.ams.org/journals/tran/1999-351-07/S0002-9947-99-02141-8/S0002-9947-99-02141-8.pdf

Abstract. We extend the original Glimm-Effros theorem for locally compact groups to a class of Polish groups including the nilpotent ones and those with an invariant metric. For this class we thereby obtain the topological Vaught conjecture. 0. Preface In this paper we consider equivalence relations induced by Polish groups acting continuously on Polish spaces. We improve on 0.1 Theorem (Sami). The topological Vaught conjecture holds for abelian Polish groups. and extend to a class of groups much wider than abelian the dichotomy theorem established by Glimm and Effros in the locally compact case. In approximate order of presentation, the main results are: 0.2 Theorem. Let G be a nilpotent Polish group acting continuously on a Polish space. Then there are either only countably many orbits or 2ℵ0 many (in fact, perfectly many). This strengthens 0.1. Let E0 be the Vitali-like equivalence relation on 2N given by xE0y ⇔∃N∈