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

TL;DR

This paper studies the existence of Schreier decorations and balanced orientations as factors of iid on various infinite transitive graphs, revealing which lattices admit such structures and their invariance properties.

Contribution

It establishes the existence of factor-of-iid Schreier decorations on certain Euclidean lattices and provides counterexamples, advancing understanding of combinatorial structures in infinite graphs.

Findings

$bZ^d$ for $d extgreater 2$ admits factor-of-iid Schreier decorations

Certain even-degree lattices have factor-of-iid balanced orientations

Existence of such structures is not invariant under quasi-isometry

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 $\mathbb{Z}^d,d\geq3$, 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.