Year: 2024
Type: preprint
Abstract: We study the free part of the Bernoulli action of $\mathbb{Z}^n$ for $n\geq 2$ and the Borel combinatorics of the associated Schreier graphs. We construct orthogonal decompositions of the spaces into marker sets with various additional properties. In general, for Borel graphs $Γ$ admitting weakly orthogonal decompositions, we show that $χ_B(Γ)\leq 2χ(Γ)-1$ under some mild assumptions. As a consequence, we deduce that the Borel chromatic number for $F(2^{\mathbb{Z}^n})$ is $3$ for all $n\geq 2$. Weakly orthogonal decompositions also give rise to Borel unlayered toast structures. We also construct orthogonal decompositions of $F(2^{\mathbb{Z}^2})$ with strong topological regularity, in particular with all atoms homeomorphic to a disk. This allows us to show that there is a Borel perfect matching for $F(2^{\mathbb{Z}^n})$ for all $n\geq 2$ and that there is a Borel lining of $F(2^{\mathbb{Z}^2})$.
Source: arXiv (Cornell University)
Authors: Gao, Su, Jackson, Steve, Krohne, Edward, Seward, Brandon
Institution: Nankai University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4391272251