NONAMENABLE SUBFORESTS OF MULTI-ENDED QUASI-PMP GRAPHS

Ruiyuan Chen, Grigory Terlov, Anush Tserunyan

correctmedium confidence

Category: Not specified
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The model’s construction mirrors the paper’s weighted cycle-cutting (w-maximal subforest) method and invokes the same sequence of results: the cycle-invariance lemma, collapsing bifurcations, dense end-maps, elimination of isolated nonvanishing ends a.e., and the Tserunyan–Tucker-Drob characterization for acyclic graphings. Minor discrepancies are present (notably a locally finite vs. locally countable slip and some theorem renumbering), but the substantive argument coincides with the paper’s and is correct under the paper’s hypotheses.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

This work establishes a clean and robust bridge between ends in quasi-pmp graphings and amenability, via a weighted cycle-cutting algorithm that produces subforests with abundant nonvanishing ends. The main theorem is natural and compelling, and the argument is technically clean and modular. The paper’s contribution includes a conceptually useful generalization of the free minimal spanning forest to the quasi-pmp/nonunimodular setting. A few minor clarifications (local finiteness where end spaces are used; citation numbering) would improve readability, but the results appear correct and significant.