Let Ξ be a countably infinite group. Given kβN, we use
Free(kΞ) to denote the free part of the Bernoulli shift action
of Ξ on kΞ. Seward and Tucker-Drob showed that there exists a
free subshift SβFree(2Ξ) such that every
free Borel action of Ξ on a Polish space admits a Borel
Ξ-equivariant map to S. Here we generalize this result as
follows. Let S be a subshift of finite type (for example,
S could be the set of all proper colorings of the Cayley graph of
Ξ with some finite number of colors). Suppose that Ο:Free(kΞ)βS is a continuous Ξ-equivariant
map and let Stab(Ο) be the set of all group elements that fix
every point in the image of Ο. Unless Ο is constant,
Stab(Ο) is a finite normal subgroup of Ξ. We prove that
there exists a subshift Sβ²βS such that the
stabilizer of every point in Sβ² is Stab(Ο) and every
free Borel action of Ξ on a Polish space admits a Borel
Ξ-equivariant map to Sβ². As an application, we deduce that
if F is a nonempty finite symmetric subset of Ξ of size β£Fβ£=d not
containing the identity and Col(F,d+1)β(d+1)Ξ is
the set of all proper (d+1)-colorings of the Cayley graph of Ξ
corresponding to F, then there is a free subshift SβCol(F,d+1) such that every free Borel action of Ξ on a Polish
space admits a Borel Ξ-equivariant map to S.Comment: 22 p