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Dual Teichmuller and lamination spaces

V. V. Fock, A. B. Goncharov

TL;DR

<3-5 sentence high-level summary>This work provides a unified, coordinate-driven treatment of the A- and X-versions of Teichmüller and lamination spaces for open (ciliated) surfaces, including extended boundary decorations. It develops explicit triangulation-based coordinates, analyzes their transformation rules under flips, and constructs canonical additive and multiplicative pairings between laminations and Teichmüller spaces, tying laminations to tropical limits of Teichmüller data. The paper also defines Poisson and degenerate symplectic structures compatible with triangulations and places these spaces in the broader context of higher Teichmüller theory and cluster varieties, with insights into combinatorial models (modular groupoids) and Markov-number phenomena. These constructions enable a constructive, elementary grasp of a rich geometric framework underlying modern Teichmüller theory and its higher-rank generalizations.

Abstract

We survey explicit coordinate descriptions for two (A and X) versions of Teichmuller and lamination spaces for open 2D surfaces, and extend them to the more general set-up of surfaces with distinguished collections of points on the boundary. Main features, such as mapping class group action, Poisson and symplectic structures and others, are described in these terms. The lamination spaces are interpreted as the tropical limits of the Teichmuller ones. Canonical pairings between lamination and Teichmuller spaces are constructed. The paper could serve as an introduction to higher Teichmuller theory developed by the authors in math.AG/0311149, math.AG/0311245.

Dual Teichmuller and lamination spaces

TL;DR

<3-5 sentence high-level summary>This work provides a unified, coordinate-driven treatment of the A- and X-versions of Teichmüller and lamination spaces for open (ciliated) surfaces, including extended boundary decorations. It develops explicit triangulation-based coordinates, analyzes their transformation rules under flips, and constructs canonical additive and multiplicative pairings between laminations and Teichmüller spaces, tying laminations to tropical limits of Teichmüller data. The paper also defines Poisson and degenerate symplectic structures compatible with triangulations and places these spaces in the broader context of higher Teichmüller theory and cluster varieties, with insights into combinatorial models (modular groupoids) and Markov-number phenomena. These constructions enable a constructive, elementary grasp of a rich geometric framework underlying modern Teichmüller theory and its higher-rank generalizations.

Abstract

We survey explicit coordinate descriptions for two (A and X) versions of Teichmuller and lamination spaces for open 2D surfaces, and extend them to the more general set-up of surfaces with distinguished collections of points on the boundary. Main features, such as mapping class group action, Poisson and symplectic structures and others, are described in these terms. The lamination spaces are interpreted as the tropical limits of the Teichmuller ones. Canonical pairings between lamination and Teichmuller spaces are constructed. The paper could serve as an introduction to higher Teichmuller theory developed by the authors in math.AG/0311149, math.AG/0311245.

Paper Structure

This paper contains 38 sections, 1 theorem, 35 equations, 11 figures.

Table of Contents

  1. Introduction.
  2. Surfaces and triangulations.
  3. Laminations.
  4. Unbounded measured laminations.
  5. Definition
  6. Construction of coordinates.
  7. Properties of the coordinates.
  8. Bounded measured laminations.
  9. Definition.
  10. Construction of coordinates.
  11. Reconstruction.
  12. Properties of the coordinates.
  13. Real lamination spaces.
  14. Teichmüller spaces.
  15. Teichmüller space of surfaces with holes ${\@fontswitch\mathcal{T}}^x(S)$.
  16. ...and 23 more sections

Key Result

Proposition 1

A square of a flip is a symmetry: if $|\Gamma_\alpha ,\Gamma|$ is a flip in an edge $\alpha$, then $|\Gamma, \Gamma_\alpha|$ is also a flip andThe notation R.n indicates the number $n$ of graphs entering this relation. R.2.$|\Gamma, \Gamma_\alpha| |\Gamma_\alpha, \Gamma|=1$. Flips in disjoint edges

Figures (11)

  • Figure 1: Examples of triangulations of ciliated surfaces. Internal edges are thick.
  • Figure 3: Twisting curves incident to a boundary component without cilia.
  • Figure 5: Gluing triangles for $\mathsf x^\alpha=2$.
  • Figure 6: Changing coordinates under a flip of the triangulation. Only the changing coordinates are shown here, the numbers on the other edges remain unchanged.
  • Figure 8: Changing of coordinates under a flip of the triangulation. Only the changing coordinates are shown here, the numbers on the other edges remain unchanged.
  • ...and 6 more figures

Theorems & Definitions (2)

  • Definition 1
  • Proposition 1