关于保型表示的Bowditch问题与Gold猜想
From Bowditch question to Goldman conjecture for type-preserving representations
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中文总结 AI 辅助
针对亏格g≥2的带孔曲面，研究映射类群在保型表示的相对PSL(2,R)特征簇上的作用，证明在特定分支上Bowditch问题的肯定回答可推出Goldman猜想。
中文摘要 AI 辅助
对于亏格g≥2的带孔曲面Σ_{g,p}，我们研究映射类群在保型表示的相对PSL(2,R)特征簇上的作用动力学。Goldman猜想预测，映射类群在这类相对特征簇的非Teichmüller分支上的作用是遍历的；相关的Bowditch问题则询问，每个非初等、非Fuchsian的保型表示是否会将某些非周边简单闭曲线映射为非双曲元。我们证明，在由周边元素像的固定符号及相对欧拉类在广义Milnor-Wood不等式中非极值所标记的相对特征簇分支上，对Bowditch问题的肯定回答可推出Goldman猜想。
英文摘要
For punctured surfaces $Σ_{g,p}$ of genus $g\geqslant 2$, we explore the dynamics of the mapping class group action on the relative $\mathrm{PSL}(2,\mathbb{R})$-character varieties of type-preserving representations. For such relative character varieties, Goldman's conjecture predicts that the mapping class group acts ergodically on their non-Teichmüller components. A related question of Bowditch asks whether every non-elementary type-preserving representation that is non-Fuchsian sends some non-peripheral simple closed curve to a non-hyperbolic element. We show that, on the components of the relative character variety indexed by fixed signs of images of peripheral elements and relative Euler classes non-extremal in the generalized Milnor-Wood inequality, an affirmative answer to Bowditch's question implies Goldman's conjecture.
