Embedding Borel graphs into grids of asymptotically optimal dimension

TL;DR

The paper proves that Borel graphs with finite subgraphs embeddable into a d-dimensional grid can be globally embedded into a Schreier graph of a free Borel action of a group with dimension proportional to d, improving previous bounds.

Contribution

It improves the embedding bounds for Borel graphs into Schreier graphs, reducing the dimension from a logarithmic function of growth rate to a linear function of the dimension d.

Findings

Borel graphs with finite subgraphs in a d-dimensional grid embed into Schreier graphs of dimension O(d).

This result strengthens previous bounds involving polynomial growth rates.

The embedding is Borel measurable, preserving the graph structure.

Abstract

Let $G$ be a Borel graph all of whose finite subgraphs embed into the $d$-dimensional grid with diagonals. We show that then $G$ itself admits a Borel embedding into the Schreier graph of a free Borel action of $\mathbb Z^{O(d)}$. This strengthens an earlier result of the authors, in which $O(d)$ is replaced by $O(\rho \log \rho)$, where $\rho$ is the polynomial growth rate of $G$.