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arXiv · 2510.23256

Invariant measures on the space of measured laminations for subgroups of mapping class group

Inhyeok Choi·Dongryul M. Kim
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Abstract

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 Hamenst\"adt'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 Teichm\"uller spaces and partially $\operatorname{CAT}(-1)$ spaces.

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Inhyeok Choi, Dongryul M. Kim. 2025-10-27. Invariant measures on the space of measured laminations for subgroups of mapping class group. https://arxiv.org/abs/2510.23256

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