Invariant measures on the space of measured laminations for subgroups of mapping class group
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918173067968512 |
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| author | Choi, Inhyeok Kim, Dongryul M. |
| author_facet | Choi, Inhyeok Kim, Dongryul M. |
| contents | For a non-elementary subgroup of the mapping class group of a surface, we study its invariant Radon measures on the space of measured laminations, by classifying them on the recurrent measured laminations. In particular, given a divergence-type subgroup, we show the uniquely ergodic by explicitly constructing the ergodic measure. This generalizes Lindenstrauss--Mirzakhani's result and Hamenstädt's result for the full mapping class group, in which case the ergodic measure is the Thurston measure. As a special case, we deduce that for a convex cocompact subgroup, every invariant ergodic Radon measure on the space of all measured laminations is either the unique measure on recurrent measured laminations, or a counting measure on the orbit of a non-recurrent measured lamination.
Our method is geometric and does not rely on continuous or homogeneous flows on the ambient space or a dynamical system associated with a finite measure space. This leads to a unifying approach for various metric spaces, including Teichmüller spaces and partially $\operatorname{CAT}(-1)$ spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_23256 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Invariant measures on the space of measured laminations for subgroups of mapping class group Choi, Inhyeok Kim, Dongryul M. Dynamical Systems Group Theory Geometric Topology For a non-elementary subgroup of the mapping class group of a surface, we study its invariant Radon measures on the space of measured laminations, by classifying them on the recurrent measured laminations. In particular, given a divergence-type subgroup, we show the uniquely ergodic by explicitly constructing the ergodic measure. This generalizes Lindenstrauss--Mirzakhani's result and Hamenstädt's result for the full mapping class group, in which case the ergodic measure is the Thurston measure. As a special case, we deduce that for a convex cocompact subgroup, every invariant ergodic Radon measure on the space of all measured laminations is either the unique measure on recurrent measured laminations, or a counting measure on the orbit of a non-recurrent measured lamination. Our method is geometric and does not rely on continuous or homogeneous flows on the ambient space or a dynamical system associated with a finite measure space. This leads to a unifying approach for various metric spaces, including Teichmüller spaces and partially $\operatorname{CAT}(-1)$ spaces. |
| title | Invariant measures on the space of measured laminations for subgroups of mapping class group |
| topic | Dynamical Systems Group Theory Geometric Topology |
| url | https://arxiv.org/abs/2510.23256 |