Canonical representations of surface groups

2022/05/30 by Aaron Landesman, Daniel Litt, Landesman, Aaron +1 · 3 citations
Mathematics · #Geometric and Algebraic Topology #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.2205.15352

Abstract

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: π₁(Σ_g,n)→ GL_r(ℂ) is a representation whose conjugacy class has finite orbit under the mapping class group, and r<√(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.

Cited by

• Rigidity of SU(2) and SO(3) quantum representations of mapping class groups at prime levels
• Global fixed point in low-dimensional surface group deformation space
• Density of integral points in the Betti moduli of quasi-projective varieties

Related

• Representations of surface groups with universally finite mapping class group orbit
• Representations of surface groups with finite mapping class group orbits
• Arithmetic representations of mapping class groups
• Linear representations of the mapping class group of dimension at most 3g-3
• Components of spaces of surface group representations