Pseudo-Anosov flows, hyperbolic geometry, and the curve graph

Starting with a pseudo-Anosov flow $φ$ on a closed hyperbolic $3$-manifold $M$ and an embedded surface $S \subset M$ that is (almost) transverse to $φ$, we relate the hyperbolic geometry of $M$ (e.g. volume, circumference, short geodesics) to dynamical invariants of $φ$ encoded by the curve graph of $S$.

💡 Research Summary

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This paper establishes quantitative links between the dynamics of a pseudo‑Anosov (or almost pseudo‑Anosov) flow φ on a closed hyperbolic 3‑manifold M and the hyperbolic geometry of M, using the curve graph C(S) of an embedded surface S that is (almost) transverse to φ. The authors generalize earlier results that were limited to the case where S is a global cross‑section (i.e., a fiber surface) to the far more general situation where S merely intersects the flow transversely but may miss some orbits.

The central objects are the stable and unstable multicurves c⁽ˢ⁾ and c⁽ᵘ⁾ obtained by intersecting the invariant stable/unstable foliations of φ with S. When S is not a fiber, each of these multicurves contains a non‑empty collection of closed leaves. The authors introduce a new invariant, the core complexity c(M ÷ S) of the cut manifold obtained by cutting M along S. Roughly, c(M ÷ S)=0 if the cut manifold contains a product annulus; otherwise it is the minimal absolute Euler characteristic of a properly embedded surface in the cut manifold that is transverse to a dynamic blow‑up of φ and meets both sides of the cut. This invariant measures how far S is from being a genuine fiber.

The main results are two theorems that relate curve‑graph distances to geometric quantities:

Theorem A (Volume and Circumference).
There exist constants k_S (depending only on |χ(S)|) and k_{M÷S} (depending on |χ(S)| and c(M ÷ S)) such that
1. d_{C(S)}(c⁽ˢ⁾,c⁽ᵘ⁾) ≥ k_S·vol(M) − k_{M÷S},
2. d_{C(S)}(c⁽ˢ⁾,c⁽ᵘ⁾) ≥ k_S·ℓ_M(γ) − k_{M÷S},

where γ is any closed geodesic intersecting S essentially and ℓ_M(γ) denotes its hyperbolic length. When a product annulus exists, c(M ÷ S)=0, so the constants depend only on the topology of S.

Theorem B (Short Geodesics).
Given any ε>0, there is a constant K=K(ε,|χ(S)|) such that if the subsurface distance d_{C(Y)}(c⁽ˢ⁾,c⁽ᵘ⁾) (for a subsurface Y⊂S) exceeds K plus a term linear in c(M ÷ S), then the geodesic β representing the curve class of Y in M satisfies ℓ_M(β)<ε. In other words, if the stable and unstable multicurves are sufficiently complicated when viewed from a distant subsurface, the corresponding geodesic in M must be arbitrarily short.

The proof proceeds in three stages. First, the authors identify curves in the cut manifold that have bounded intersection with c⁽ˢ⁾ and c⁽ᵘ⁾. When a product annulus is present these are the components of Thurston’s “window frame”; otherwise they are the boundary components of a minimal‑complexity surface. Second, they show that these curves can be realized as hyperbolic geodesics of uniformly bounded length in M. This uses a “broken windows” construction together with the Masur–Minsky distance‑to‑length estimates for curve graphs. Third, they pass to the quasi‑Fuchsian cover associated to S and apply standard Kleinian group techniques to replace c⁽ˢ⁾ and c⁽ᵘ⁾ by nearby bounded‑length curves, thereby obtaining the desired volume, circumference, and short‑geodesic bounds.

The paper also discusses several examples and applications. Section 7 treats depth‑one and finite‑depth foliations, showing how the theorems specialize in those contexts. It presents explicit constructions (via dynamic blow‑ups) that realize prescribed curve‑graph distances and core complexities, and it revisits Whitfield’s examples where “junctures” can be made arbitrarily short, illustrating why the authors must work with curves intersecting the multicurves rather than the multicurves themselves.

Overall, the work extends the deep interplay between hyperbolic 3‑manifold geometry, pseudo‑Anosov dynamics, and combinatorial topology of surface curve graphs beyond the fibered case. It provides a robust framework for quantifying how the “complexity” of the stable/unstable multicurves forces global geometric features of the ambient manifold, and it opens new directions for studying non‑fibered transverse surfaces, core complexity bounds, and their implications for volume, systolic geometry, and the structure of Kleinian groups.