Factor-of-iid Schreier decorations of lattices in Euclidean spaces

Abstract

A Schreier decoration is a combinatorial coding of an action of the free group $F_d$ 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 ${\mathbb{Z}}^d,d\ge 3$, 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.

25 pages. This is an extended version of the first part of the earlier preprint "Factor of iid Schreier decorations of transitive graphs''. Non-amenable results are split off into a separate preprint titled "Factor-of-iid balanced orientation of non-amenable graphs''

References in corpus (2)

• Percolation on dual lattices with k-fold symmetry
• Factor-of-iid balanced orientation of non-amenable graphs