2403.11326
Borel Complexity of the Isomorphism Relation of Archimedean Orders in Finitely Generated Groups
Antoine Poulin
correctmedium confidence
- Category
- math.DS
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves that E(GL_n(Z) ↷ Ar(Z^n)) is not hyperfinite for n ≥ 3 and not treeable for n ≥ 4 via a GL_n(Z)-equivariant identification Ar(Z^n) ≅ (R^n)*_ti / R_{>0} (Proposition 2.5) and a measure-theoretic route using amenability and property (T), culminating in Theorems 2.22 and 2.23; the candidate solution reaches the same conclusions using a ping–pong construction of free subgroups and inheritance properties of hyperfiniteness/treeability. The model’s proof needs minor justification for standard facts it cites (e.g., that a free Borel action of a non-amenable group cannot yield a hyperfinite orbit relation), but the conclusions match the paper’s results. See Proposition 2.5, Lemma 2.21, Corollary 2.15, Proposition 2.20, and Theorems 2.22–2.23 in the paper .
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The paper gives concise, correct proofs establishing sharp thresholds for hyperfiniteness and treeability in the orbit-equivalence relations arising from GL\_n(Z) acting on Archimedean orders, thereby answering a question raised in the literature. The method—transport via a class-bijective identification to a co-null subset of R\^n, then applying Zimmer amenability and property (T) tools—is standard yet effective. Minor expository enhancements would improve readability, but the core arguments are sound and valuable for researchers in descriptive set theory and ordered groups.