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arxiv: 2605.25082 · v1 · pith:ZLNP7ZVWnew · submitted 2026-05-24 · 🧮 math.DS · math.GT

Exotic codimension one Anosov flows

Sergio Fenley , Kathryn Mann , Rafael Potrie This is my paper

Pith reviewed 2026-06-29 23:47 UTC · model grok-4.3

classification 🧮 math.DS math.GT
keywords Anosov flowscircle bundleshyperbolic 3-manifoldsCannon-Thurston mapspseudo-Anosov flowsVerjovsky conjectureorbit equivalence4-manifolds
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The pith

Circle bundles over closed hyperbolic 3-manifolds carry Anosov flows, giving counterexamples to Verjovsky's conjecture and manifolds with infinitely many such flows up to orbit equivalence.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs Anosov flows on certain 4-manifolds formed as circle bundles over closed hyperbolic 3-manifolds. These examples refute a conjecture of Verjovsky that no Anosov flows exist on such bundles. The construction proceeds by lifting pseudo-Anosov quasigeodesic flows from the base 3-manifolds via their associated Cannon-Thurston maps. Some of the resulting 4-manifolds support infinitely many distinct Anosov flows up to orbit equivalence. A sympathetic reader would care because the result shows that the space of Anosov flows on 4-manifolds is larger and more flexible than earlier expectations allowed.

Core claim

We construct Anosov flows in certain circle bundles over closed hyperbolic 3-manifolds, producing counterexamples to a conjecture of Verjovsky. Some of these 4-manifolds admit infinitely many distinct Anosov flows up to orbit equivalence. The construction is made by using Cannon-Thurston maps associated to pseudo-Anosov quasigeodesic flows in hyperbolic 3-manifolds.

What carries the argument

Cannon-Thurston maps associated to pseudo-Anosov quasigeodesic flows in hyperbolic 3-manifolds, which lift the base dynamics to produce the Anosov flow on the circle bundle.

If this is right

  • Certain circle bundles over closed hyperbolic 3-manifolds admit Anosov flows.
  • These flows furnish counterexamples to Verjovsky's conjecture.
  • Some of the 4-manifolds support infinitely many Anosov flows up to orbit equivalence.
  • The Cannon-Thurston maps from pseudo-Anosov quasigeodesic flows supply the mechanism that produces the flows.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same lifting technique may extend to other fiber bundles whose base admits pseudo-Anosov flows with Cannon-Thurston maps.
  • Orbit-equivalence classes of Anosov flows on 4-manifolds may be uncountable in more cases than previously known.
  • The existence result constrains attempts to classify Anosov flows by their underlying manifold topology.

Load-bearing premise

The Cannon-Thurston maps coming from pseudo-Anosov quasigeodesic flows in the base 3-manifolds can be used to define Anosov flows on the total space of the circle bundle.

What would settle it

An explicit computation on one of the constructed bundles showing that the lifted vector field fails to be Anosov, or a proof that no such lift exists for any choice of bundle and base flow.

Figures

Figures reproduced from arXiv: 2605.25082 by Kathryn Mann, Rafael Potrie, Sergio Fenley.

Figure 1
Figure 1. Figure 1: The special case f = id, dim(M) = 2. On the left, some leaves Mf× {p} of F h with geodesics pointing towards f(p), the graph of f : S 1 → ∂Mf is the orange line. On the right, a leaf of F v consisting of geodesics from ξ to f(p) at each Mf × {p}. We now define a 2-dimensional foliation F v in Mf × S 1 transverse to F h . For each ξ ∈ ∂Mf, let Lξ denote the union (over all p ∈ S 1 ) of leaves of G with nega… view at source ↗
Figure 2
Figure 2. Figure 2: Transversals and first-returns along αe. In the left one sees the configuration leading to a contradiction and in the right the actual configura￾tion. Let T ′ ⊂ T be the subrectangle containing α ∩ T where the first-return to T along ϕ t -orbits is defined and continuous. We choose T small enough so that the union of orbit segments from T ′ to T is contained in the tubular neighborhood Bα. Let r : T ′ → T … view at source ↗
read the original abstract

We construct Anosov flows in certain circle bundles over closed hyperbolic 3-manifolds, producing counterexamples to a conjecture of Verjovsky. Some of these 4-manifolds admit infinitely many distinct Anosov flows up to orbit equivalence. The construction is made by using Cannon-Thurston maps associated to pseudo-Anosov quasigeodesic flows in hyperbolic $3$-manifolds.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The paper constructs Anosov flows on certain circle bundles over closed hyperbolic 3-manifolds by using Cannon-Thurston maps associated to pseudo-Anosov quasigeodesic flows. This produces counterexamples to a conjecture of Verjovsky, and shows that some of these 4-manifolds admit infinitely many distinct Anosov flows up to orbit equivalence.

Significance. If the construction is correct, the result is significant: it supplies explicit counterexamples to Verjovsky's conjecture in the setting of codimension-one Anosov flows and demonstrates that certain 4-manifolds carry infinitely many orbit-inequivalent Anosov flows. The approach via Cannon-Thurston maps from 3-manifold dynamics provides a concrete method for producing such examples.

minor comments (1)
  1. [Abstract] The abstract states the main results clearly but does not indicate the precise topological conditions on the circle bundles or the 3-manifolds beyond hyperbolicity; a short clarifying sentence would help readers locate the examples.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance.

Circularity Check

0 steps flagged

No significant circularity; construction relies on external Cannon-Thurston maps

full rationale

The paper's central claim is an explicit construction of Anosov flows on circle bundles over hyperbolic 3-manifolds, using Cannon-Thurston maps associated to pseudo-Anosov quasigeodesic flows. This relies on established prior results in 3-manifold topology (Cannon-Thurston maps are classical, not defined here). No equations or steps in the abstract reduce a prediction or uniqueness claim to a fitted input or self-citation chain. The argument is presented as building new flows from known maps and foliations, with details supplied in the full text for hyperbolicity estimates. This matches the default expectation of a non-circular construction paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review is abstract-only; the ledger is therefore minimal and provisional. The central claim rests on domain-standard facts about Cannon-Thurston maps whose details are not supplied here.

axioms (1)
  • domain assumption Cannon-Thurston maps exist and have the required properties for pseudo-Anosov quasigeodesic flows on hyperbolic 3-manifolds
    Invoked explicitly as the method of construction in the abstract.

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