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Filtered screens and augmented Teichmüller space

Douglas J. LaFountain, R. C. Penner

TL;DR

The paper introduces filtered screens to bordify the decorated Teichmüller space $\tilde{T}(F)$ and identifies when paths yield short curves, connecting local simplicial data to global hyperbolic degeneration. It constructs a decorated augmented Teichmüller space $\widehat{T}(F)$ via a quotient of the filtered-screen space and provides a mapping-class-group–invariant CW decomposition indexed by partially oriented stratum graphs, with strata as products of (partially decorated) Teichmüller spaces across irreducible components. A central technical tool is the Filtered IJ Lemma, which relates growth/vanishing rates of lambda lengths to quasi-recurrent subgraphs, enabling precise detection of pinch curves. The framework unifies combinatorial fatgraph models with the geometry of augmented Teichmüller space, offering new ways to study the homotopy type and potential cycles in the Deligne–Mumford compactification via a combinatorial, filtered-screens approach.

Abstract

We study a new bordification of the decorated Teichmüller space for a multiply punctured surface F by a space of filtered screens on the surface that arises from a natural elaboration of earlier work of McShane-Penner. We identify necessary and sufficient conditions for paths in this space of filtered screens to yield short curves having vanishing length in the underlying surface F. As a result, an appropriate quotient of this space of filtered screens on F yields a decorated augmented Teichmüller space which is shown to admit a CW decomposition that naturally projects to the augmented Teichmüller space by forgetting decorations and whose strata are indexed by a new object termed partially oriented stratum graphs.

Filtered screens and augmented Teichmüller space

TL;DR

The paper introduces filtered screens to bordify the decorated Teichmüller space and identifies when paths yield short curves, connecting local simplicial data to global hyperbolic degeneration. It constructs a decorated augmented Teichmüller space via a quotient of the filtered-screen space and provides a mapping-class-group–invariant CW decomposition indexed by partially oriented stratum graphs, with strata as products of (partially decorated) Teichmüller spaces across irreducible components. A central technical tool is the Filtered IJ Lemma, which relates growth/vanishing rates of lambda lengths to quasi-recurrent subgraphs, enabling precise detection of pinch curves. The framework unifies combinatorial fatgraph models with the geometry of augmented Teichmüller space, offering new ways to study the homotopy type and potential cycles in the Deligne–Mumford compactification via a combinatorial, filtered-screens approach.

Abstract

We study a new bordification of the decorated Teichmüller space for a multiply punctured surface F by a space of filtered screens on the surface that arises from a natural elaboration of earlier work of McShane-Penner. We identify necessary and sufficient conditions for paths in this space of filtered screens to yield short curves having vanishing length in the underlying surface F. As a result, an appropriate quotient of this space of filtered screens on F yields a decorated augmented Teichmüller space which is shown to admit a CW decomposition that naturally projects to the augmented Teichmüller space by forgetting decorations and whose strata are indexed by a new object termed partially oriented stratum graphs.

Paper Structure

This paper contains 18 sections, 19 theorems, 42 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Background and notation
  3. Decorated Teichmüller Spaces
  4. Coordinates and Parameters
  5. Dual punctured fatgraphs
  6. Convex Hull Construction and Cell Decomposition
  7. Space of filtered screens
  8. Screens
  9. Filtered screens
  10. The space of filtered screens
  11. Stable paths and the Filtered IJ Lemma
  12. Stable paths and filtered screens
  13. Decorated augmented Teichmüller space
  14. Augmented Teichmüller space
  15. Punctured fatgraphs with partial pairing
  16. ...and 3 more sections

Key Result

Theorem 2.4

[Theorem 2.2.5 of [P3]] For any quasi triangulation $E$ of $F$ based at a set ${\mathcal{P}}\neq\emptyset$ of punctures, the natural mapping given by is a real-analytic homeomorphism.

Figures (15)

  • Figure 1: h-lengths and simplicial coordinates.
  • Figure 2: Standard notation near an edge $e$.
  • Figure 3: A screen $\{A, A', E\}$ along with its boundary.
  • Figure 4: A subfatgraph of $G(E)$.
  • Figure 5: Filtered screen blow up of cell decomposition for $F_0^3$.
  • ...and 10 more figures

Theorems & Definitions (54)

  • Definition 2.1: Decorated and undecorated punctures, $\widetilde{T}_{\mathcal{P}}(F)$
  • Definition 2.2: Ideal arc, quasi cell decomposition
  • Definition 2.3: Lambda lengths
  • Theorem 2.4
  • Definition 2.5: h-lengths
  • Definition 2.6: Simplicial coordinates
  • Definition 2.7: Punctured fatgraph and vertices
  • Definition 2.8: Boundary component of a punctured fatgraph
  • Definition 2.9: Quasi efficient, simple cycles and horocycles
  • Lemma 2.10: Telescoping Lemma
  • ...and 44 more