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

Ruiyuan Chen, Antoine Poulin, Ran Tao, Anush Tserunyan

TL;DR

This work proves several results showing that every locally finite Borel graph whose large-scale geometry is"tree-like"induces a treeable equivalence relation, and shows treeability of countable Borel equivalence relations equipped with an abstract wallspace structure on each class obeying some local finiteness conditions, which is called a proper walling.

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.

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

TL;DR

This work proves several results showing that every locally finite Borel graph whose large-scale geometry is"tree-like"induces a treeable equivalence relation, and shows treeability of countable Borel equivalence relations equipped with an abstract wallspace structure on each class obeying some local finiteness conditions, which is called a proper walling.

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.
Paper Structure (25 sections, 85 theorems, 42 equations, 6 figures)

This paper contains 25 sections, 85 theorems, 42 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Notions of tree-like graphs
  3. Dense families of cuts
  4. Wallspaces and median graphs
  5. The case of one end
  6. Remark on Borel versus countable combinatorics
  7. Preliminaries
  8. Graphs
  9. Median graphs
  10. Hyperplanes and half-spaces
  11. Pocsets and duality
  12. Finiteness conditions on median graphs and pocsets
  13. Subpocsets and wallspaces
  14. Ends
  15. Quasi-isometries and coarse equivalences
  16. ...and 10 more sections

Key Result

theorem 1

If a CBER admits a locally finite Borel graphing whose components are quasi-trees, then it is Borel treeable.

Figures (6)

  • Figure 1: The process of converting "tree-like" structures into trees. Number on arrow refers to result(s) showing that step for a countable structure. Number on box refers to end result on treeability of CBERs.
  • Figure 2: A dual median graph with a half-space $H$ with no principal orientations (circled vertices) on its boundary, and such that the set $S$ of principal orientations in $H$ closest to its boundary do not uniquely identify it (the left side of the vertical hyperplane down the middle would have the same $S$).
  • Figure 3: Shrinking a neighborhood $A$ of an end $U$ to a connected-coconnected subneighborhood.
  • Figure 4: The sides $H$ of the parts $F^{-1}(y)$ of a tree decomposition $F$ are dense towards ends.
  • Figure 5: Pruning "leaf half-spaces" from a one-ended median graph to get a one-ended tree. The thick highlighted edges form the $4$th equivalence relation $F_4$ in the resulting witness to hyperfiniteness.
  • ...and 1 more figures

Theorems & Definitions (197)

  • theorem 1: \ref{['thm:cber-quasitreeable-treeable']}
  • theorem 2: \ref{['thm:cber-quasitreeable-bddeg']}
  • remark 1
  • theorem 3: \ref{['thm:cber-bddtw-treeable']}
  • theorem 4: \ref{['thm:cber-cuts-dense-treeable']}
  • theorem 5: \ref{['thm:cber-wallable-treeable']}
  • theorem 6: \ref{['thm:cber-median']}
  • theorem 7: \ref{['thm:cber-quasitreeing-bddtw-end-hypf']}
  • theorem 8: \ref{['thm:cber-median-end-hypf']}
  • theorem 9: \ref{['thm:med-subtree']}
  • ...and 187 more