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Flexibility and degeneracy around a theorem of Thurston

Alexander Nolte

TL;DR

The paper investigates degenerate and flexible phenomena around Thurston's asymmetric metric by constructing geodesics robust to ε-Lipschitz perturbations and by producing open sets of multi-surface representations whose limit cones are explicitly polyhedral. It introduces a non-rotating length correction that isolates twisting effects and proves uniform distortion bounds, enabling precise control over length data across pinched geometries. A second major contribution describes how limit cones for sums of Fuchsian representations can be forced to be cones over finite-sided polyhedra, even when the ambient group is infinitely generated, illustrating remarkable degeneracy and local non-rigidity. Together, these results shed light on the generic structure around Thurston’s theorem, revealing both analytic flexibility in Thurston geodesics and combinatorial tractability in limit-cone descriptions. The work contributes new understanding of the local behavior of Thurston's metric and the geometry of representation spaces with potential broader applications in Teichmüller theory and higher-rank dynamics.

Abstract

We give two flexible and degenerate constructions related to a theorem of Thurston. First, we produce geodesic segments for Thurston's asymmetric metric on Teichmüller space $\mathcal{T}(S_g)$ that remain geodesics after adding arbitrary $\varepsilon$-Lipschitz noise to all but one Fenchel-Nielsen coordinate. Then, for all $2 < n \leq 3g-3$ we construct open sets in $\mathcal{T}(S_g)^n$ for which the limit cones of the corresponding representations in $\mathrm{PSL}_2(\mathbb{R})^n$ are cones over explicit finite-sided polyhedra. Each construction is as degenerate as possible and has applications to the basic structure and local non-rigidity of the involved objects.

Flexibility and degeneracy around a theorem of Thurston

TL;DR

The paper investigates degenerate and flexible phenomena around Thurston's asymmetric metric by constructing geodesics robust to ε-Lipschitz perturbations and by producing open sets of multi-surface representations whose limit cones are explicitly polyhedral. It introduces a non-rotating length correction that isolates twisting effects and proves uniform distortion bounds, enabling precise control over length data across pinched geometries. A second major contribution describes how limit cones for sums of Fuchsian representations can be forced to be cones over finite-sided polyhedra, even when the ambient group is infinitely generated, illustrating remarkable degeneracy and local non-rigidity. Together, these results shed light on the generic structure around Thurston’s theorem, revealing both analytic flexibility in Thurston geodesics and combinatorial tractability in limit-cone descriptions. The work contributes new understanding of the local behavior of Thurston's metric and the geometry of representation spaces with potential broader applications in Teichmüller theory and higher-rank dynamics.

Abstract

We give two flexible and degenerate constructions related to a theorem of Thurston. First, we produce geodesic segments for Thurston's asymmetric metric on Teichmüller space that remain geodesics after adding arbitrary -Lipschitz noise to all but one Fenchel-Nielsen coordinate. Then, for all we construct open sets in for which the limit cones of the corresponding representations in are cones over explicit finite-sided polyhedra. Each construction is as degenerate as possible and has applications to the basic structure and local non-rigidity of the involved objects.

Paper Structure

This paper contains 53 sections, 57 theorems, 49 equations, 4 figures.

Table of Contents

  1. Thurston Geodesics
  2. Limit Cones
  3. Non-rotating length
  4. Further applications on Thurston geodesics
  5. Remarks on Proofs
  6. Thurston Geodesics
  7. Limit Cones
  8. Outline
  9. Conventions and Fundamentals
  10. Notational Conventions
  11. Annuli and Collars
  12. Fenchel-Nielsen Coordinates
  13. Lipschitz maps
  14. Gluing
  15. Sums of Fuchsian Representations
  16. ...and 38 more sections

Key Result

Theorem 1

On the complement of a closed set of Hausdorff codimension $1$ in $\mathcal{T}(S) \times \mathcal{T}(S)$, the supremum $\mathrm{K}(X_1, X_2)$ is realized by a simple curve $\gamma$ that remains optimal for all pairs in a neighborhood of $(X_1, X_2)$.

Figures (4)

  • Figure 1: Left: a genus two surface, with a convenient pants decomposition in purple, seams in red, and a teal closed curve $\gamma$. Right: lifts of $\gamma$ and seams curves to $\mathcal{A}(\gamma_1)$ for the middle pants curve $\gamma_1$ that demonstrates the spiraling in the middle contributes to the combinatorial rotation $r(\gamma, \gamma_1)$.
  • Figure 2: Labels on a hexagon and one of the pentagons it is split into.
  • Figure 3: The basic set-up after lifting to the universal cover.
  • Figure 4: Twisting in a cylinder and its lift to $\mathbb H^2$.

Theorems & Definitions (116)

  • Theorem 1: Thurston, thurston1998minimal
  • Remark 1.1
  • Corollary 2: $\mathrm{L}^\infty$ regions
  • Theorem 3: Noisy Geodesics
  • Corollary 4: Geodesic Germs Non-Rigid
  • Theorem 5: Polyhedral Limit Cones
  • Remark 1.2
  • Theorem 6
  • Corollary 7: Not Locally Rigid
  • Definition 1.3: Rotation number
  • ...and 106 more