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Electric Teichmüller spaces and $k$-multicurve graphs

Kento Sakai

Abstract

Masur and Minsky showed that the curve graph is quasi-isometric to the Teichmüller space electrified along its thin part, and hence the Teichmüller space is weakly relatively hyperbolic with respect to the thin part. In this paper, we extend this result to the $k$-multicurve graph by electrifying the Teichmüller space along the thin part where the extremal length of $k$ curves is sufficiently small. A key ingredient is a bound on the $k$-multicurve graph distance in terms of the intersection number, which is obtained by adapting the upper bound for the pants graph due to Lackenby and Yazdi.

Electric Teichmüller spaces and $k$-multicurve graphs

Abstract

Masur and Minsky showed that the curve graph is quasi-isometric to the Teichmüller space electrified along its thin part, and hence the Teichmüller space is weakly relatively hyperbolic with respect to the thin part. In this paper, we extend this result to the -multicurve graph by electrifying the Teichmüller space along the thin part where the extremal length of curves is sufficiently small. A key ingredient is a bound on the -multicurve graph distance in terms of the intersection number, which is obtained by adapting the upper bound for the pants graph due to Lackenby and Yazdi.
Paper Structure (12 sections, 25 theorems, 48 equations, 6 figures, 1 table)

This paper contains 12 sections, 25 theorems, 48 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Metric geometry
  4. Multicurve graphs
  5. Quadratic differentials and measured foliations
  6. Extremal length and Teichmüller space
  7. Length of curves
  8. Inequality between intersection numbers and distance
  9. Inequality for pants graph distance
  10. Inequality of $k$-multicurve graph distance
  11. Quasi-isometry between $\hat{\mathcal{T}}^k$ and $\mathcal{C}^{[k]}$
  12. Quadratic upper bound for pants graph distance via intersection number

Key Result

Theorem 1

The electrified Teichmüller space $\hat{\mathcal{T}}^{k}(\Sigma)$ is quasi-isometric to the $k$-multicurve graph $\mathcal{C}^{[k]}(\Sigma)$.

Figures (6)

  • Figure 1:
  • Figure 2:
  • Figure 3: The red and blue arcs denote $e_j$ and $e_{j+1}$ connecting boundary components of $N$.
  • Figure 4:
  • Figure 5: The white region (bounded by $\partial N_{j+1}$ and the black dotted line) is $S$.
  • ...and 1 more figures

Theorems & Definitions (39)

  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Corollary 4
  • Theorem 5
  • Definition 2.1
  • Theorem 2.2: Kerckhoff's formula kerckhoff1980asymptotic
  • Theorem 2.3: bers1985inequality
  • Corollary 2.4
  • Theorem 2.5: maskit1985comparison
  • ...and 29 more