映射类群子群作用的Hopf分解
Hopf decomposition of the actions of subgroups of the mapping class group
- School of Mathematics and Physics, College of Science and Engineering, Kanazawa University（金泽大学理工学学术院数理物理学系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过Kaimanovich准则和Dirichlet多面体刻画了Teichmüller模群子群在Thurston边界上作用的Hopf分解，并证明了Torelli群的大horospherical极限集具有全测度。
AI中文摘要：
我们研究了Teichmüller模群子群在Thurston边界上关于Thurston测度类的作用的Hopf分解。Kaimanovich的一般Radon--Nikodym准则使我们能够通过由极值长度表示的级数的发散与收敛来描述保守部分和耗散部分。我们将保守部分与（模去零集后的）大horospherical极限集等同。对于子群中具有平凡稳定子的任意基点，我们将耗散部分（模去零集）与Dirichlet点的集合以及相关Dirichlet多面体理想边界的子群平移之并等同。通过Dirichlet多面体的描述利用了Teichmüller空间不同点处极值长度比水平集具有零测度这一事实。对于亏格至少为二的闭曲面的Torelli群，我们利用周期映射的径向极限证明其锥形极限集具有零测度。将我们的几何刻画与Choi、Gekhtman、Yang和Zheng建立的其边界作用的保守性相结合，我们得到其大horospherical极限集具有全测度，且每个Dirichlet多面体的理想边界具有零测度。
英文摘要：
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.
