English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We prove several results showing that every locally finite Borel graph whoselarge-scale geometry is "tree-like" induces a treeable equivalence relation. Inparticular, our hypotheses hold if each component of the original graph eitherhas bounded tree-width or is quasi-isometric to a tree, answering a question ofTucker-Drob. In the latter case, we moreover show that there exists a Borelquasi-isometry to a Borel forest, under the additional assumption of(componentwise) bounded degree. We also extend these results on quasi-treeingsto Borel proper metric spaces. In fact, our most general result showstreeability of countable Borel equivalence relations equipped with an abstractwallspace structure on each class obeying some local finiteness conditions,which we call a proper walling. The proof is based on the Stone duality betweenproper wallings and median graphs, i.e., CAT(0) cube complexes. Finally, westrengthen the conclusion of treeability in these results to hyperfiniteness inthe 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