Treeable Graphings Are Local Limits of Finite Graphs

Abstract

Let $\mathbf{G}$ be a graphing, that is a Borel graph defined by $d$ measure preserving involutions. We prove that if $\mathbf{G}$ is {\em treeable} then it arises as the local limit of some sequence $(G_n{)}_{n\in \mathbb{N}}$ of graphs with maximum degree at most $d$. This extends a result by Elek [G. Elek, Note on limits of finite graphs, Combinatorica 27 (2007)] (for $\mathbf{G}$ a treeing) and consequently extends the domain of the graphings for which Aldous-Lyons conjecture is known to be true.