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A geometric boundary for the moduli space of grafted surfaces

Andrea Egidio Monti

Abstract

Let $S$ be a closed orientable surface of genus at least two. We introduce a bordification of the moduli space $\mathcal{PT}(S)$ of complex projective structures, with a boundary consisting of projective classes of half-translation surfaces. Thurston established an equivalence between complex projective structures and hyperbolic surfaces grafted along a measured lamination, leading to a homeomorphism $\mathcal{PT}(S) \cong \mathcal{T}(S) \times \mathcal{ML}(S)$. Our bordification is geometric in the sense that convergence to points on the boundary corresponds to the geometric convergence of grafted surfaces to half-translation surfaces (up to rescaling). This result relies on recent work by Calderon and Farre on the orthogeodesic foliation construction. Finally, we introduce a change of perspective, viewing grafted surfaces as a deformation (which we term "inflation") of half-translation surfaces, consisting of inserting negatively curved regions.

A geometric boundary for the moduli space of grafted surfaces

Abstract

Let be a closed orientable surface of genus at least two. We introduce a bordification of the moduli space of complex projective structures, with a boundary consisting of projective classes of half-translation surfaces. Thurston established an equivalence between complex projective structures and hyperbolic surfaces grafted along a measured lamination, leading to a homeomorphism . Our bordification is geometric in the sense that convergence to points on the boundary corresponds to the geometric convergence of grafted surfaces to half-translation surfaces (up to rescaling). This result relies on recent work by Calderon and Farre on the orthogeodesic foliation construction. Finally, we introduce a change of perspective, viewing grafted surfaces as a deformation (which we term "inflation") of half-translation surfaces, consisting of inserting negatively curved regions.

Paper Structure

This paper contains 8 sections, 3 theorems, 4 equations.

Table of Contents

  1. Grafting
  2. Main results
  3. Further work and future directions
  4. Outline of the paper
  5. Acknowledgment
  6. Preface
  7. Teichmüller space
  8. Half-translation surfaces and quadratic differentials

Key Result

Theorem A

Inside $\mathop{\mathrm{MMet}}\nolimits(S)$, $\mathcal{PT}(S)$ is asymptotic to the cone given by $\mathcal{QT}(S)/S^1$, or in other words, after projecting to the real projectification $\mathbb{P}\mathop{\mathrm{MMet}}\nolimits(S)$, we obtain

Theorems & Definitions (3)

  • Theorem A
  • Theorem B
  • Theorem C