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2205.02495

ORBITAL EQUIVALENCE CLASSES OF FINITE COVERINGS OF GEODESIC FLOWS

THIERRY BARBOT, SÉRGIO R. FENLEY

correcthigh confidence
Category
Not specified
Journal tier
Specialist/Solid
Processed
Sep 28, 2025, 12:56 AM

Audit review

The paper’s Main Theorem states exactly the classification claimed by the model: if the fiberwise index n is odd there is a single orbital equivalence class of Anosov flows on M, and if n is even there are exactly two classes. The proof in the paper reduces the problem via Ghys’ theorem to finite fiberwise covers of geodesic flows, identifies the parameter space Gn of covers with an affine H1(Σ; Z/n)-space, and computes Mod±(Σ)-orbits using Lickorish generators, yielding the parity dichotomy (Theorem 3.23) . The model’s solution follows the same outline: Ghys reduction , parameterization by subgroups Γ ⊂ π1(T1Σ) (Definition 3.16) , the orbital-equivalence ⇔ Mod±(Σ)-orbit correspondence (Proposition 3.19) , and the orbit computation via Dehn twists (core of Proposition 3.22, used in the proof of Theorem 3.23) . Minor imprecisions in the model include writing Mod(Σ) instead of Mod±(Σ) and saying the action factors through Sp(2g, Z) rather than the correct Sp(2g, Z/nZ) on H1(Σ; Z/n) (Remark 3.18) , but these do not affect the stated classification, which matches the paper’s result and argument.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The work gives a clean, correct classification of orbital equivalence classes of Anosov flows on circle bundles covering T\^1Σ, revealing a parity-based dichotomy. The argument is largely topological, reducing via Ghys’ theorem and then computing mapping-class-group orbits on an affine H\^1(Σ;Z/n)-space using Lickorish generators. Minor clarifications about the extended mapping class group and the affine (not linear) nature of the action would strengthen clarity.