Canonical representations of surface groups

• 成果类型: Article
• 署名作者: Landesman, Aaron; Litt, Daniel
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2024.199.2.6
• 发表日期: 2024
• 页码: 823-897
• 关键词: mapping class group; algebraic-solutions; MODULI SPACES; linear representations; differential-equations; topological dynamics; local systems; higgs bundles; finite covers; painleve

摘要

Let Sigma(g,n) be an orientable surface of genus g with n punctures. We study actions of the mapping class group Mod(g,n) of Sigma(g,n) via Hodge -theoretic and arithmetic techniques. We show that if rho : pi(1)(Sigma(g,n)) -> GL(r)(C) is a representation whose conjugacy class has finite orbit under Mod(g,n), and r < root g + 1, then rho 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.