We show that for any infinite countable group  G    and for any free Borel action   G↷X     G  ↷  X        there exists an equivariant class-bijective Borel map from  X    to the free part   Free(2 G )      Free   (    2    G    )        of the 2-shift   G↷2 G      G  ↷    2    G         . This implies that any Borel structurability which holds for the equivalence relation generated by   G↷Free(2 G )     G  ↷   Free   (    2    G    )        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(2 G )      Free   (    2    G    )        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   2 G        2    G          lands in the free part. As a corollary we obtain that for every   ϵ>0     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.
['Brandon Seward', 'Robin D. Tucker-Drob']
Borel structurability on the 2-shift of a countable group