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Bending parameterization of one-sided degenerated Kleinian surface groups

Bruno Dular

TL;DR

We address the problem of extending bending-parameterization rigidity to one-sided degenerations of Kleinian surface groups. The main result shows that for one-sided degenerated structures $\rho\in{\mathcal AH}_o^+(S;e^-)$, the manifold is uniquely determined by the end structure $\mathrm E^-(\rho)$ and the top bending lamination $\mathrm b^+(\rho)$, i.e. by the pair $(m^-(\rho), b^+(\rho))$ up to the specified end data. The authors prove this by establishing continuity, properness, and injectivity of the map $\mathrm m^-\\times\\mathrm b^+$ from ${\mathcal AH}_o^+(S;e^-)$ to $\mathcal T(\Sigma^-)\\times{\mathcal ML}^{\perp e^-}_{<\pi}(S^+)$, leveraging the ending lamination theorem, Thurston’s grafting parameterization, and Morgan–Shalen limits. Consequently, the work extends the bending-parameterization paradigm from quasi-Fuchsian to a broad class of degenerations and clarifies how end invariants control the geometry of the convex core and conformal boundary.

Abstract

It was recently proved that quasi-Fuchsian manifolds are uniquely determined by their bending laminations. This paper concerns a similar result for certain non-quasi-Fuchsian manifolds~: those obtained by degenerating one end but not the other, i.e. those appearing in boundaries of Bers slices. More precisely, we show that such hyperbolic manifolds are uniquely determined by the end structure of the degenerated end and the bending lamination of the other. The end structure consists of the parabolic locus, which is a multicurve, together with ending laminations or conformal structures on each components of its complement.

Bending parameterization of one-sided degenerated Kleinian surface groups

TL;DR

We address the problem of extending bending-parameterization rigidity to one-sided degenerations of Kleinian surface groups. The main result shows that for one-sided degenerated structures , the manifold is uniquely determined by the end structure and the top bending lamination , i.e. by the pair up to the specified end data. The authors prove this by establishing continuity, properness, and injectivity of the map from to , leveraging the ending lamination theorem, Thurston’s grafting parameterization, and Morgan–Shalen limits. Consequently, the work extends the bending-parameterization paradigm from quasi-Fuchsian to a broad class of degenerations and clarifies how end invariants control the geometry of the convex core and conformal boundary.

Abstract

It was recently proved that quasi-Fuchsian manifolds are uniquely determined by their bending laminations. This paper concerns a similar result for certain non-quasi-Fuchsian manifolds~: those obtained by degenerating one end but not the other, i.e. those appearing in boundaries of Bers slices. More precisely, we show that such hyperbolic manifolds are uniquely determined by the end structure of the degenerated end and the bending lamination of the other. The end structure consists of the parabolic locus, which is a multicurve, together with ending laminations or conformal structures on each components of its complement.
Paper Structure (21 sections, 18 theorems, 19 equations, 1 figure)

This paper contains 21 sections, 18 theorems, 19 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Outline of the proof
  3. Further directions
  4. Background
  5. Laminations and trees
  6. Pleated surfaces
  7. End structures
  8. Metrics converging to an end structure
  9. Quasi-conformal deformations of Kleinian groups
  10. Continuity
  11. Holonomy of projective structures
  12. Thurston's grafting parameterization of projective structures
  13. One-sided degenerated structures
  14. Properness
  15. Algebraic convergence
  16. ...and 6 more sections

Key Result

Theorem 1

Consider the map $\mathop{\mathrm{m}}\nolimits^-\times\mathop{\mathrm{b}}\nolimits^+\colon{\mathcal{QF}}_o(S)\to{\mathcal{T}}(S^-)\times{\mathcal{ML}}_{<\pi}(S^+)$ sending a quasi-Fuchsian structure $\rho$ to the pair $(\mathop{\mathrm{m}}\nolimits^-(\rho),\mathop{\mathrm{b}}\nolimits^+(\rho))$ of t

Figures (1)

  • Figure 1: Schematic representation of hyperbolic manifolds homeomorphic to $S\times{\mathbb{R}}$, with $1$-dimensional "$S$-direction", conformal boundary in red and convex core in grey. The structures are, from left to right: Fuchsian, quasi-Fuchsian, Kleinian with a cusp and Kleinian with simply-degenerated bottom end.

Theorems & Definitions (28)

  • Theorem : Theorem \ref{['thm:prescribed_metric_and_bending']} in the text
  • Theorem
  • Remark 2.1
  • Theorem 2.3: Morgan-Shalen MR769158MR1339846MR1402300
  • Theorem 2.5: Bonahon MR1413855
  • Theorem 2.6: Brock MR1791139
  • Theorem 2.11
  • Theorem 2.12: Ending lamination theorem, Brock-Minsky-Canary MR2925381
  • Corollary 3.5
  • proof
  • ...and 18 more