论文标题
表面组的规范表示
Canonical representations of surface groups
论文作者
论文摘要
令$σ_{g，n} $为带有$ n $ dumctures的属$ g $的定向表面。我们通过Hodge理论和算术技术研究$σ_{G，N} $的映射类组的动作。我们表明，如果$$ρ：π_1（σ_{g，n}）\ to gl_r（\ mathbb {c}）$$是一种表示，其共轭类在映射类组下具有有限的轨道，而$ r <\ sqrt {g+1} $，则是$ρ$，然后$ρ$有限的图像。这回答了Junho Peter Whang和Mark Kisin的问题。我们将方法应用于Putman-Wieland的猜想，Fontaine-Mazur猜想以及Esnault-Kerz的问题。 这些证据依赖于非亚伯式霍奇理论，我们先前关于异词性变形的准确性的工作以及Esnault-Groechenig和Klevdal-Patrikis在Simpson在同胞上刚性刚性本地系统上的完整性猜想的最新工作。
Let $Σ_{g,n}$ be an orientable surface of genus $g$ with $n$ punctures. We study actions of the mapping class group of $Σ_{g,n}$ via Hodge-theoretic and arithmetic techniques. We show that if $$ρ: π_1(Σ_{g,n})\to GL_r(\mathbb{C})$$ is a representation whose conjugacy class has finite orbit under the mapping class group, and $r<\sqrt{g+1}$, then $ρ$ has finite image. This answers questions of Junho Peter Whang and Mark Kisin. We give applications of our methods to the Putman-Wieland conjecture, the Fontaine-Mazur conjecture, and a question of Esnault-Kerz. The proofs rely on non-abelian Hodge theory, our earlier work on semistability of isomonodromic deformations, and recent work of Esnault-Groechenig and Klevdal-Patrikis on Simpson's integrality conjecture for cohomologically rigid local systems.
