Representations of surface groups with universally finite mapping class group orbit

2019/07/09 by Lawrence, Brian, Litt, Daniel
#Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Topology (math.GT) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1907.03941

Abstract

Let Σ_g,n be the orientable genus g surface with n punctures, where 2-2g-n<0. Let ρ: π₁(Σ_g,n)→ GL_m(ℂ) be a representation. Suppose that for each finite covering map f: Σ_{g', n'}→ Σ_{g, n}, the orbit of (the isomorphism class of) f^*(ρ) under the mapping class group MCG(Σ_g',n') of Σ_g',n' is finite. Then we show that ρ has finite image. The result is motivated by the Grothendieck-Katz p-curvature conjecture, and gives a reformulation of the p-curvature conjecture in terms of isomonodromy.

Related

• Canonical representations of surface groups
• Representations of surface groups with finite mapping class group orbits
• Arithmetic representations of mapping class groups
• Geometric realizations of cyclic actions on surfaces
• Linear representations of the mapping class group of dimension at most 3g-3