Higher dimensional obstructions for star reductions

Abstract

A $\ast$-reduction between two equivalence relations is a Baire measurable reduction which preserves generic notions, i.e., preimages of meager sets are meager. We show that a $\ast$-reduction between orbit equivalence relations induces generically an embedding between the associated Becker graphs. We introduce a notion of dimension for Polish $G$-spaces which is generically preserved under $\ast$-reductions. For every natural number $n$ we define a free action of $S_{\infty }$ whose dimension is $n$ on every invariant Baire measurable non-meager set. We also show that the $S_{\infty }$-space which induces the equivalence relation ${=}^{+}$ of countable sets of reals is $\infty$-dimensional on every invariant Baire measurable non-meager set. We conclude that the orbit equivalence relations associated to all these actions are pairwise incomparable with respect to $\ast$-reductions.

20 pages, 2 figures. Final version

References in corpus (1)

• Generic representations of abelian groups and extreme amenability