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Vaught's conjecture on analytic sets

Author: Greg Hjorth
Journal: J. Amer. Math. Soc. 14 (2001), 125-143
MSC (2000): Primary 03E15
Published electronically: September 18, 2000
MathSciNet review: 1800351
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Let $G$ be a Polish group. We characterize when there is a Polish space $X$ with a continuous $G$-action and an analytic set (that is, the Borel image of some Borel set in some Polish space) $A\subset X$ having uncountably many orbits but no perfect set of orbit inequivalent points.

Such a Polish $G$-space $X$ and analytic $A$ exist exactly when there is a continuous, surjective homomorphism from a closed subgroup of $G$ onto the infinite symmetric group, $S_\infty$, consisting of all permutations of $\mathbb{N} $ equipped with the topology of pointwise convergence.

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Additional Information

Greg Hjorth
Affiliation: Department of Mathematics, University of California, Los Angeles, California 90095-1555

Keywords: Polish group, group actions, topological Vaught conjecture
Received by editor(s): June 8, 1998
Received by editor(s) in revised form: June 22, 2000
Published electronically: September 18, 2000
Additional Notes: The author’s research was partially supported by NSF grant DMS 96-22977.
Article copyright: © Copyright 2000 American Mathematical Society

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