In this talk we will consider *-reductions between orbit equivalence relations. These are Baire measurable reductions which preserve generic notions, i.e., preimages of comeager sets are comeager. In short, *-reductions are weaker than Borel reductions in the sense of definability, but as we will see, they are much more sensitive to the dynamics of the orbit equivalence relations in question.

Based on a past joint work with M. Lupini we will introduce a notion of dimension for Polish G-spaces. This dimension is always 0 whenever the group G admits a complete and left invariant metric, but in principle, it can take any value n within 0,1,....∞ For each such n we will produce a free action of S_∞ which is generically n-dimensional and we will deduce that the associated orbit equivalence relations are pairwise incomparable with respect to *-reductions.

This is in joint work with A. Kruckman.