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Quasi-treeings are treeable: a streamlined proof

Zhaoshen Zhai

TL;DR

The paper provides a streamlined proof that a countable Borel equivalence relation $E$ generated by a locally-finite graphing with components that are quasi-trees is treeable. It constructs dense, finitely-separating families of cuts within each component, forms the dual median graph from these cuts, and derives canonical spanning trees via a Borel cycle-cutting approach. By encoding per-component trees into a standard Borel space and using principal orientations, the authors obtain a global Borel treeing of the graphing, yielding a treeable $E$. A generalization is given: any locally-finite graphing with a Borel, dense-in-ends family of cuts also yields a treeable relation, clarifying the link between wall-like cuts, median graphs, and treeings.

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.

Quasi-treeings are treeable: a streamlined proof

TL;DR

The paper provides a streamlined proof that a countable Borel equivalence relation generated by a locally-finite graphing with components that are quasi-trees is treeable. It constructs dense, finitely-separating families of cuts within each component, forms the dual median graph from these cuts, and derives canonical spanning trees via a Borel cycle-cutting approach. By encoding per-component trees into a standard Borel space and using principal orientations, the authors obtain a global Borel treeing of the graphing, yielding a treeable . A generalization is given: any locally-finite graphing with a Borel, dense-in-ends family of cuts also yields a treeable relation, clarifying the link between wall-like cuts, median graphs, and treeings.

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.
Paper Structure (10 sections, 26 theorems, 15 equations)

This paper contains 10 sections, 26 theorems, 15 equations.

Table of Contents

  1. Preliminaries on Pocsets, Ends of Graphs, and Median Graphs
  2. Profinite pocsets of cuts
  3. Ends of graphs
  4. Median graphs and projections
  5. Convex half-spaces of median graphs
  6. Graphs with Dense Families of Cuts
  7. The Dual Median Graph of a Profinite Pocset and its Spanning Trees
  8. Construction of the dual median graph
  9. Canonical spanning trees
  10. Borel Treeings of Graphings with Dense Cuts

Key Result

Theorem A

If a CBER $E$ admits a locally-finite graphing such that each component is a quasi-tree,Recall that metric spaces $X$ and $Y$ are quasi-isometric if they are isometric up to a bounded multiplicative and additive error, and $X$ is a quasi-tree if it is quasi-isometric to a simplicial tree; see Gro93

Theorems & Definitions (63)

  • Theorem A: Section \ref{['sec:borel_treeings_of_graphings_with_dense_cuts']}, CPTT23*Theorem 1.1
  • Theorem B: Propositions \ref{['prp:construction_of_dual_median_graph']}, \ref{['prp:dual_median_graph_of_cuts_has_finite_hyperplanes']}, \ref{['prp:canonical_spanning_trees']}
  • Definition 1.1
  • Remark
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • Definition 1.4
  • Definition 1.5
  • ...and 53 more