Hopf decomposition of the actions of subgroups of the mapping class group

Abstract

We study the Hopf decomposition of subgroup actions of the Teichm\"uller modular group on the Thurston boundary with respect to the Thurston measure class. Kaimanovich's general Radon--Nikodym criteria allow us to describe the conservative and dissipative parts by the divergence and convergence of a series expressed in terms of extremal length. We identify the conservative part with the big horospherical limit set modulo null sets. For any basepoint with trivial stabilizer in the subgroup, we identify the dissipative part, modulo null sets, with the set of Dirichlet points and with the union of the subgroup translates of the ideal boundary of the associated Dirichlet polyhedron. The description via Dirichlet polyhedra uses the fact that level sets of extremal-length ratios at distinct points of Teichm\"uller space have measure zero. 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. Combining our geometric characterization with the conservativity of its boundary action established by Choi, Gekhtman, Yang, and Zheng, we obtain that its big horospherical limit set has full measure and that the ideal boundary of each Dirichlet polyhedron has measure zero.