Applications of tree decompositions and accessibility to treeability of Borel graphs

TL;DR

This paper introduces a framework for tree decompositions of Borel graph components, establishing accessibility and treeability criteria, and applies these to show measure treeability of certain Borel equivalence relations, extending prior results.

Contribution

It develops a Borel framework for tree decompositions and accessibility, providing new criteria for treeability of Borel graphs and equivalence relations, generalizing previous work.

Findings

Borel equivalence relations with accessible planar components are measure treeable.

Uniformly locally finite Borel graphs with finite tree-width components are Borel treeable.

Certain property (T) equivalence relations do not admit planar graphings almost surely.

Abstract

A framework to handle tree decompositions of the components of a Borel graph in a Borel fashion is introduced, along the lines of Tserunyan's Stallings Theorem for equivalence relations arXiv:1805.09506. This setting leads to a notion of accessibility for Borel graphs, together with a treeability criterion. This criterion is applied to show that, in particular, Borel equivalence relations associated to Borel graphs with accessible planar connected components are measure treeable, generalising results of Conley, Gaboriau, Marks, and Tucker-Drob arXiv:2104.07431 and Timar arXiv:1910.01307. It is also proven that uniformly locally finite Borel graphs with components of finite tree-width yield Borel treeable equivalence relations. Our results imply that p.m.p countable Borel equivalence relations with measured property (T) do not admit locally finite graphings with planar components a.s.