Quasi-treeings are treeable: a streamlined proof

Abstract

We present a streamlined exposition of a construction by R. Chen, A. Poulin, R. Tao, and A. Tserunyan, which proves the treeability of equivalence relations generated by any locally-finite Borel graph such that each component is a quasi-tree. More generally, we show that if each component of a locally-finite Borel graph admits a finitely-separating Borel family of cuts, then we may 'canonically' replace each component of the graph by a tree of special ultrafilter-like objects on cuts called orientations; moreover, if the cuts are dense towards ends, then the union of these trees is a Borel treeing.