Borel dimension growth and hyperfiniteness

Abstract

We show that every increasing union of Borel graphs of pointwise volume growth at most $\exp (O(r^{\gamma }))$ is hyperfinite, where $\gamma \approx 0.1523$ is the unique real root of $(1-\gamma {)}^3-4\gamma =0$. This implies a positive answer to the special case of Weiss's question on the hyperfiniteness of Borel actions of countable amenable groups, for countable amenable groups locally of volume growth $\exp (O(r^{\gamma }))$ for $\gamma$ as above. We also show that every bounded degree Borel graph of subexponential volume growth has Borel Følner tilings, improving a result of Downarowicz and Zhang for graphs generated by free Borel actions of groups of subexponential volume growth. Our main tools are the development of a Borel analogue of the two-parameter dimension growth function of a metric space as introduced by Dranishnikov and Sapir, and ball carving algorithms from theoretical computer science. We end with a pair of conjectures relating Borel amenability, Borel subexponential dimension growth, and hyperfiniteness, which would imply a positive answer to Weiss's question.