Hexagon decompositions and the Weil-Petersson metric
Abstract
We give an explicit quasi-isometry from the flip graph of triangulations ℱg,n to the ε-thick part of Teichmüller space equipped with the Teichmüller metric, by mapping each ideal triangulation to the hyperbolic surface whose shearing coordinates along that triangulation are all equal to zero. Extending this construction, we obtain a quasi-isometry Q from the hexagon graph ℋg,n to the augmented Teichmüller space Teich̅g,n equipped with the Weil-Petersson metric, and we estimate its width, that is, the Hausdorff distance between Q(ℋg,n) and Teich̅g,n. By bounding the Weil-Petersson distance along grafting rays and the Teichmüller distance along shearing deformations, we show that the width is at most √(g+n)log(g+n). We also provide an explicit projection from any point in Teich̅g,n to the thick part of boundary strata, maintaining simultaneous control over the Weil-Petersson distance and the shearing coordinates.
Related
- Generalized Hitchin-Kobayashi correspondence and Weil-Petersson current
- Closed Response Calculus for SLE Weldings and Weil--Petersson Kähler Geometry
- Scalar curvature, volume, and Weil-Petersson translation length
- The Weil-Petersson diameter of moduli space for large genus
- Eynard-Orantin Topological Recursions for Higher Weil-Petersson Volumes on Moduli Spaces of Super Riemann Surfaces
BibTeX
@misc{abadie2026hexagon,
title = {{Hexagon decompositions and the Weil-Petersson metric}},
author = {Marie Abadie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31278},
eprint = {2609.31278},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31278},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Hexagon decompositions and the Weil-Petersson metric
AU - Marie Abadie
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31278
UR - https://arxiv.org/abs/2609.31278
AB - We give an explicit quasi-isometry from the flip graph of triangulations ℱ_g,n to the ε-thick part of Teichmüller space equipped with the Teichmüller metric, by mapping each ideal triangulation to the hyperbolic surface whose shearing coordinates along that triangulation are all equal to zero. Extending this construction, we obtain a quasi-isometry Q from the hexagon graph ℋ_g,n to the augmented Teichmüller space Teich̅_g,n equipped with the Weil-Petersson metric, and we estimate its width, that is, the Hausdorff distance between Q(ℋ_g,n) and Teich̅_g,n. By bounding the Weil-Petersson distance along grafting rays and the Teichmüller distance along shearing deformations, we show that the width is at most √(g+n)log(g+n). We also provide an explicit projection from any point in Teich̅_g,n to the thick part of boundary strata, maintaining simultaneous control over the Weil-Petersson distance and the shearing coordinates.
ER -