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The descriptive set theory of Polish group actions. (English) Zbl 0949.54052
London Mathematical Society Lecture Note Series. 232. Cambridge: Cambridge University Press. xi, 136 p. (1996).
The structure of Borel actions of Polish (locally compact) groups has long been studied in ergodic theory, operator algebra, and group representation theory. The book under review contains the results of the authors in the theory of definable actions of Polish groups (not necessarily locally compact) and the associated orbit equivalence relations. In particular, actions of the symmetric group $$S_\infty$$, the group of homeomorphisms $$H(X)$$ of a compact metric space $$X$$, the unitary group $$U(H)$$ of a separable infinite dimensional Hilbert space $$H$$, etc., are in the field of interest. However, some of the general results about actions of Polish groups proved in the book are new even in the locally compact case. The following six major themes are studied in the book:
(1) How to change the topology of a space $$X$$ on which a Borel action of a Polish group becomes continuous in the new topology. The main result here says that every Borel action on a Polish space is Borel isomorphic to a continuous one. Another result says that for a continuous action of a Polish group $$G$$ on a Polish space $$X$$ and any invariant Borel set $$B\subset X$$ there is a finer Polish topology on $$X$$ with the same Borel structure in which $$B$$ is clopen and the action remains continuous.
(2) The existence of universal actions is the second theme. The authors prove that for every Polish group $$G$$ there is a universal Borel $$G$$-space $$\mathcal U_G$$ which can be moreover a compact Polish $$G$$-space. This means that every Borel $$G$$-space can be embedded into $$\mathcal U_G$$ via a Borel isomorphism. These results have applications to Tarski’s theorem on paradoxical decompositions, namely the authors prove this optimal analog of Tarski’s theorem: For a Borel action of a Polish group $$G$$ on a Polish space $$X$$ there is a $$G$$-invariant probability Borel measure if and only if there is no countable $$G$$-paradoxical decomposition of $$X$$.
(3) The logic actions which are in fact actions of a group $$S_\infty$$ over countable structures of countable relational languages. The orbit equivalence relations are isomorphisms between these structures. It is shown that they are concrete examples of the universal $$S_\infty$$-action.
(4) The next theme is the theme of dichotomies for the orbit space. One example of such dichotomy is the Vaught conjecture and the other important dichotomies discussed in the book are the Silver dichotomy and the dichotomies of the Glimm-Effros type. There are variants of the Topological Vaught Conjecture which assert that there are either countably many or perfectly many orbits in any continuous (or Borel) action of a Polish group on a Polish space (or on a Borel invariant set). The authors prove that various forms of the Topological Vaught Conjecture are equivalent. Moreover, they prove that the Topological Vaught Conjecture for the symmetric group $$S_\infty$$ is equivalent to the usual model-theoretic Vaught Conjecture for the language $$L_{\omega_1\omega}$$.
(5) The question of descriptive complexity for the orbit equivalence relation induced by a Borel action is very natural. It is shown that such an equivalence relation is in general analytic but not Borel. Anyway, every Borel $$G$$-space can still be decomposed into $$\aleph_1$$ many Borel invariant sets. The fact whether the orbit equivalence relation is Borel depends on the fact whether the stabilizer function for the group $$G$$ is Borel.
(6) The last theme is the theme of definable cardinality of the orbit space of an action which is in fact a generalization of the theme on dichotomies. Most results here are obtained assuming the axiom of determinacy for games on reals.

##### MSC:
 54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) 03E15 Descriptive set theory 54-02 Research exposition (monographs, survey articles) pertaining to general topology 03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations 28A05 Classes of sets (Borel fields, $$\sigma$$-rings, etc.), measurable sets, Suslin sets, analytic sets 54E50 Complete metric spaces 54H11 Topological groups (topological aspects)