A Schreier decoration is a combinatorial coding of an action of the free group Fd
on the vertex set of a 2d-regular graph. We investigate whether a Schreier decoration
exists on various countably infinite transitive graphs as a factor of iid.
We show that Zd,d≥3, the square lattice and also the three other Archimedean lattices
of even degree have finitary-factor-of-iid Schreier decorations, and exhibit examples
of transitive graphs of arbitrary even degree in which obtaining such a decoration
as a factor of iid is impossible.
We also prove that symmetrical planar lattices with all degrees even have a factor
of iid balanced orientation, meaning the indegree of every vertex is equal to its
outdegree, and demonstrate that the property of having a factor-of-iid balanced orientation
is not invariant under quasi-isometry.
