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Effective length-projection bounds for hyperbolic 3-manifolds diffeomorphic to $S\times\mathbb{R}$

Gabriele Viaggi

TL;DR

This paper proves an explicit, computable relation between the end-invariant data of a hyperbolic 3-manifold $Q\simeq S\times\mathbb{R}$ and the lengths of geodesic representatives of subsurface boundary multicurves. Leveraging end-invariant projections $d_Y(\nu^-,\nu^+)$ and the curve graph, the author derives concrete bounds for $\ell_Q(\partial Y)$ in terms of $d_Y(\nu^-,\nu^+)$ with explicit constants, via a sequence of effective ingredients: an effective efficiency bound, an effective convex-core width bound, and a controlled hyperbolic Dehn filling step. The work builds on Minsky’s framework for non-projective projections and extends it to computable bounds, with applications to closed hyperbolic 3-manifolds fibering over the circle, producing a geometric counterpart to uniform projection bounds in fibered faces. Overall, the results provide a fully quantitative bridge between end invariants, subsurface projections, and geodesic lengths, enabling explicit numerical estimates in concrete hyperbolic 3-manifold constructions.

Abstract

We give a formula with explicit constants relating the subsurface projection $d_Y(ν^-,ν^+)$ of the end invariants $ν^-,ν^+$ of a hyperbolic 3-manifold $Q$ diffeomorphic to $S\times\mathbb{R}$ and the length of the geodesic representative in $Q$ of the multicurve $\partial Y$. This makes effective and computable the large projections versus short curves relation proved by Minsky. We give an application to closed hyperbolic 3-manifolds fibering over the circle providing a geometric analog of the uniform projection bound in fibered faces of Minsky and Taylor.

Effective length-projection bounds for hyperbolic 3-manifolds diffeomorphic to $S\times\mathbb{R}$

TL;DR

This paper proves an explicit, computable relation between the end-invariant data of a hyperbolic 3-manifold and the lengths of geodesic representatives of subsurface boundary multicurves. Leveraging end-invariant projections and the curve graph, the author derives concrete bounds for in terms of with explicit constants, via a sequence of effective ingredients: an effective efficiency bound, an effective convex-core width bound, and a controlled hyperbolic Dehn filling step. The work builds on Minsky’s framework for non-projective projections and extends it to computable bounds, with applications to closed hyperbolic 3-manifolds fibering over the circle, producing a geometric counterpart to uniform projection bounds in fibered faces. Overall, the results provide a fully quantitative bridge between end invariants, subsurface projections, and geodesic lengths, enabling explicit numerical estimates in concrete hyperbolic 3-manifold constructions.

Abstract

We give a formula with explicit constants relating the subsurface projection of the end invariants of a hyperbolic 3-manifold diffeomorphic to and the length of the geodesic representative in of the multicurve . This makes effective and computable the large projections versus short curves relation proved by Minsky. We give an application to closed hyperbolic 3-manifolds fibering over the circle providing a geometric analog of the uniform projection bound in fibered faces of Minsky and Taylor.

Paper Structure

This paper contains 40 sections, 33 theorems, 108 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Geometry and topology of hyperbolic surfaces
  3. Thick-thin decomposition
  4. Moderate-length curves
  5. Recurrence and Anosov closing
  6. Curve graphs
  7. Curve graph
  8. Distance and intersections
  9. Subsurface projections
  10. Laminations and Gromov boundary
  11. Subsurface projections for laminations
  12. Geometry and topology of 3-manifolds
  13. 3-dimensional topology
  14. Tubes and cusps
  15. Hyperbolic 3-manifolds and Kleinian groups
  16. ...and 25 more sections

Key Result

Theorem 1

There exist constants $a,b,c>0$ such that the following holds. Let $S$ be a closed orientable surface of genus at least 2. Consider a hyperbolic 3-manifold $Q$ diffeomorphic to $S\times \mathbb{R}$ with end invariants $\nu^-,\nu^+$. Then, for every proper essential connected non-annular subsurface $ then $\gamma_Y\times\{0\}$ is isotopic to a geodesic link in $Q$ of length

Figures (3)

  • Figure 1: Ideal concatenation.
  • Figure 2: Ideal pleated annulus. Non-parabolic case.
  • Figure 3: Shortcuts. If the segment $h(\xi_A)$ is homotopic relative to the endpoints to $f(\xi_Y)$ we can use the disk $D$ to remove at least one of the intersections $\gamma_Y\cap\alpha$ with a homotopy of $\alpha$.

Theorems & Definitions (77)

  • Theorem 1
  • Theorem 2
  • proof : Proof of Theorem \ref{['thm:main2']}
  • Definition 2.1: Standard Collar and Cuspidal Part
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 67 more