Polish Group Actions and the Vaught Conjecture

Abstract

We consider the topological Vaught conjecture: If a Polish group $G$ acts continuously on a Polish space $S$, then $S$ has either countably many or perfectly many orbits. We show 1. The conjecture is true for Abelian groups. 2. The conjecture is true whenever $G$, $S$ are recursively presented, the action of $G$ is recursive and, for $x \in S$ the orbit of $x$ is of Borel multiplicative rank $\leq \omega _1^x$. Assertion $1$ holds also for analytic $S$. Specializing $G$ to a closed subgroup of $\omega !$, we prove that nonempty invariant Borel sets, not having perfectly many orbits, have orbits of about the same Borel rank. An upper bound is derived for the Borel rank of orbits when the set of orbits is finite.