Borel Combinatorics of Abelian Group Actions

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 $F(2^{\mathbb{Z}^2})$.

Figures (38)

• Figure 1: Avoiding corners of $R$ and points of $S$
• Figure 2: The first two steps of the subdivision algorithm
• Figure 3: The assignment algorithm in dimension 2. The red rectangles are assigned to $R_1$, and the blue ones to $R_2$.
• Figure 4: The construction of $R"$ from $R'$. The blue rectangles are to be added and the red ones to be removed.
• Figure 5: The second construction in dimension 2.

Theorems & Definitions (94)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Definition 2.4
• Theorem 2.5