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

DOI10.1016/J.DISC.2024.114056arXiv2101.12577MaRDI QIDQ6359363

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Publication date: 29 January 2021

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 $mathbb{Z}^d,dgeq3$, 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.

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