Tree-like graphings, wallings, and median graphings of equivalence relations

TL;DR

This paper establishes that certain locally finite Borel graphs with tree-like large-scale geometry induce treeable equivalence relations, extending to proper metric spaces and using wallspace structures linked to median graphs.

Contribution

It introduces new conditions under which Borel graphs with tree-like properties are treeable, including cases with bounded tree-width, quasi-isometry to trees, and proper wallings, generalizing previous results.

Findings

Locally finite Borel graphs with tree-like geometry induce treeable equivalence relations.

Results extend to Borel proper metric spaces and wallspace structures.

Treeability can be strengthened to hyperfiniteness under certain conditions.

Abstract

We prove several results showing that every locally finite Borel graph whose large-scale geometry is "tree-like" induces a treeable equivalence relation. In particular, our hypotheses hold if each component of the original graph either has bounded tree-width or is quasi-isometric to a tree, answering a question of Tucker-Drob. In the latter case, we moreover show that there exists a Borel quasi-isometry to a Borel forest, under the additional assumption of (componentwise) bounded degree. We also extend these results on quasi-treeings to Borel proper metric spaces. In fact, our most general result shows treeability of countable Borel equivalence relations equipped with an abstract wallspace structure on each class obeying some local finiteness conditions, which we call a proper walling. The proof is based on the Stone duality between proper wallings and median graphs, i.e., CAT(0) cube complexes. Finally, we strengthen the conclusion of treeability in these results to hyperfiniteness in the case where the original graph has one (selected) end per component, generalizing the same result for trees due to Dougherty--Jackson--Kechris.