Hopf decomposition of the actions of subgroups of the mapping class group
Abstract
We study the Hopf decomposition of subgroup actions of the Teichmüller modular group on the Thurston boundary with respect to the Thurston measure class. We identify the conservative part with the big horospherical limit set modulo null sets. For a basepoint with trivial stabilizer in the subgroup, the ideal boundary of the associated Dirichlet polyhedron is wandering, and its subgroup translates cover the dissipative part modulo null sets. The dissipative part also agrees with the set of Dirichlet points modulo null sets. The description using Dirichlet polyhedra relies on a separation theorem for extremal length: every level set of an extremal length ratio at distinct points of Teichmüller space has measure zero. Kaimanovich's Radon--Nikodym criterion characterizes the two parts by the divergence and convergence, respectively, of a series of extremal length ratios. For the Torelli group of a closed surface of genus at least two, we use radial limits of the period map to prove that its conical limit set has measure zero. Together with the conservativity established by Choi, Gekhtman, Yang, and Zheng, our geometric characterization implies that its big horospherical limit set has full measure and that the ideal boundary of every Dirichlet polyhedron has measure zero.
Related
- Sphere decomposition for finite subgroups of 3-manifold mapping class groups
- Mapping class group actions on the cohomology of configuration spaces of surfaces
- Group actions on von Neumann algebras with compact open subgroups
- Baumslag-Solitar Subgroups Obstruct Model Companions for Fields with Group Actions
- Ivanov's finite-index Frattini conjecture for subgroups of mapping class groups
BibTeX
@misc{miyachi2026hopf,
title = {{Hopf decomposition of the actions of subgroups of the mapping class group}},
author = {Hideki Miyachi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.10175},
eprint = {2609.10175},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10175},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hopf decomposition of the actions of subgroups of the mapping class group
AU - Hideki Miyachi
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2609.10175
UR - https://arxiv.org/abs/2609.10175
AB - We study the Hopf decomposition of subgroup actions of the Teichmüller modular group on the Thurston boundary with respect to the Thurston measure class. We identify the conservative part with the big horospherical limit set modulo null sets. For a basepoint with trivial stabilizer in the subgroup, the ideal boundary of the associated Dirichlet polyhedron is wandering, and its subgroup translates cover the dissipative part modulo null sets. The dissipative part also agrees with the set of Dirichlet points modulo null sets. The description using Dirichlet polyhedra relies on a separation theorem for extremal length: every level set of an extremal length ratio at distinct points of Teichmüller space has measure zero. Kaimanovich's Radon--Nikodym criterion characterizes the two parts by the divergence and convergence, respectively, of a series of extremal length ratios. For the Torelli group of a closed surface of genus at least two, we use radial limits of the period map to prove that its conical limit set has measure zero. Together with the conservativity established by Choi, Gekhtman, Yang, and Zheng, our geometric characterization implies that its big horospherical limit set has full measure and that the ideal boundary of every Dirichlet polyhedron has measure zero.
ER -