Borel structurability on the 2-shift of a countable group

Abstract

We show that for any infinite countable group G and for any free Borel action G↷X there exists an equivariant class-bijective Borel map from X to the free part Free(2G) of the 2-shift G↷2G. This implies that any Borel structurability which holds for the equivalence relation generated by G↷Free(2G) must hold a fortiori for all equivalence relations coming from free Borel actions of G. A related consequence is that the Borel chromatic number of Free(2G) is the maximum among Borel chromatic numbers of free actions of G. This answers a question of Marks. Our construction is flexible and, using an appropriate notion of genericity, we are able to show that in fact the generic G-equivariant map to 2G lands in the free part. As a corollary we obtain that for every ϵ>0, every free p.m.p. action of G has a free factor which admits a 2-piece generating partition with Shannon entropy less than ϵ. This generalizes a result of Danilenko and Park.