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

TL;DR

The work develops a quantitative link between dynamics of almost pseudo-Anosov flows on closed hyperbolic 3-manifolds and hyperbolic geometry via the curve graph of a transverse surface S. By isolating stable/unstable multicurves c^s and c^u and bounding their interactions with bounded-length representatives, the authors reduce global geometric information to Cantor-set–type curve-graph data and controlled subsurface projections. Central contributions include volume and circumference bounds d_{𝒞(S)}(c^s,c^u) ≼ vol(M) + D and d_{𝒞(S)}(c^s,c^u) ≼ ℓ(γ) + D, as well as short-curve results derived from subsurface distances, all achieved by a three-stage strategy: topological reduction to window-annulus data, geometric control via JSJ and pleated-surface theory, and passage to a quasi-Fuchsian cover where standard tools yield the final inequalities. The paper also demonstrates applications to finite-depth foliations and constructs endperiodic examples showing unbounded c^s–c^u distance, thereby illustrating both the reach and limitations of the approach. Overall, the results illuminate how dynamical complexity on S governs ambient hyperbolic geometry in M through the curve-graph framework, extending beyond fibered settings to more general transverse configurations.

Abstract

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

Figures (6)

• Figure 1: Figure from landry2023endperiodic demonstrating a dynamic blowup of a 3-pronged singular orbit and its transverse cross section. Here there is a single blowup annulus.
• Figure 2: An unstable periodic half-leaf $A^u_\omega$ based at an orbit $\omega$ in $N$. It intersects $S$ in three closed leaves $\{c_1,c_2,c_3\} \subset c^u$, where $c_1$ is principal and the others cobound product flow annuli in $N$.
• Figure 3: A disk of contact in $\Sigma \cup \partial N$.
• Figure 4: Adjusting $\Sigma_L$ to include $\beta_L$.
• Figure 5: A figure from landry2023endperiodic showing a periodic leaf of $\Lambda^+$ that intersects $\partial_+ N$ in a closed curve (a principal unstable curve), and the periodic escaping ray of $\lambda^+$ coming from its intersection with $L$.

Theorems & Definitions (55)

• Theorem A: Volume and circumference
• Theorem B: Short curves
• Remark 1.1
• Lemma 2.1
• proof
• Remark 2.2
• Lemma 3.1
• proof
• Lemma 3.2: Product annuli
• proof