Effective Mapping Class Group Dynamics II: Geometric Intersection Numbers
- 1Department of Mathematics, Rice University, Houston, USA
Abstract
We show that the action of the mapping class group on the space of closed curves of a closed surface effectively tracks the corresponding action on Teichmüller space in the following sense: for all but quantitatively few mapping classes, the information of how a mapping class moves a given point of Teichmüller space determines, up to a power saving error term, how it changes the geometric intersection numbers of a given closed curve with respect to arbitrary geodesic currents. Applications include an effective estimate describing the speed of convergence of Teichmüller geodesic rays to the boundary at infinity of Teichmüller space, an effective estimate comparing the Teichmüller and Thurston metrics along mapping class group orbits of Teichmüller space, and, in the sequel and forthcoming work of Honaryar, effective estimates for countings of closed geodesics on closed, negatively curved surfaces. Furthermore, in forthcoming work of Arana-Herrera and Honaryar, the main result of this paper is applied to study the arithmetic/homological complexity of long simple closed geodesics on negatively curved surfaces.
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BibTeX
@article{aranaherrera2026effective,
title = {{Effective Mapping Class Group Dynamics II: Geometric Intersection Numbers}},
author = {Francisco Arana–Herrera},
journal = {Geometric and Functional Analysis},
year = {2026},
month = jun,
doi = {10.1007/s00039-026-00740-6},
url = {https://doi.org/10.1007/s00039-026-00740-6},
note = {CC BY-NC-ND},
}
RIS
TY - JOUR
TI - Effective Mapping Class Group Dynamics II: Geometric Intersection Numbers
AU - Francisco Arana–Herrera
PY - 2026
DA - 2026/06/01
JO - Geometric and Functional Analysis
DO - 10.1007/s00039-026-00740-6
UR - https://doi.org/10.1007/s00039-026-00740-6
AB - We show that the action of the mapping class group on the space of closed curves of a closed surface effectively tracks the corresponding action on Teichmüller space in the following sense: for all but quantitatively few mapping classes, the information of how a mapping class moves a given point of Teichmüller space determines, up to a power saving error term, how it changes the geometric intersection numbers of a given closed curve with respect to arbitrary geodesic currents. Applications include an effective estimate describing the speed of convergence of Teichmüller geodesic rays to the boundary at infinity of Teichmüller space, an effective estimate comparing the Teichmüller and Thurston metrics along mapping class group orbits of Teichmüller space, and, in the sequel and forthcoming work of Honaryar, effective estimates for countings of closed geodesics on closed, negatively curved surfaces. Furthermore, in forthcoming work of Arana-Herrera and Honaryar, the main result of this paper is applied to study the arithmetic/homological complexity of long simple closed geodesics on negatively curved surfaces.
SN - 1016-443X
SN - 1420-8970
ER -