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Canonical representations of surface groups

Aaron Landesman, Daniel Litt

TL;DR

The paper proves that any MCG-finite representation ρ: π_1(Σ_{g,n}) → GL_r(ℂ) with rank r < √(g+1) has finite image, using a synthesis of non-abelian Hodge theory, deformation theory, and cohomological vanishing results. It develops a unitary- and PVHS-based strategy to deduce integrality and finiteness, then extends the result to semisimple and non-semisimple cases via Putman–Wieland asymptotics. The work yields arithmetic consequences, including instances of the Fontaine–Mazur conjecture and nonliftable residual representations, and provides asymptotic PW results that connect to questions in geometric topology. Together, these results offer a robust framework linking MCG dynamics, Hodge theory, and arithmetic geometry, with broad implications for isomonodromic deformations and the structure of local systems on moduli spaces.

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 $$ρ: π_1(Σ_{g,n})\to GL_r(\mathbb{C})$$ 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.

Canonical representations of surface groups

TL;DR

The paper proves that any MCG-finite representation ρ: π_1(Σ_{g,n}) → GL_r(ℂ) with rank r < √(g+1) has finite image, using a synthesis of non-abelian Hodge theory, deformation theory, and cohomological vanishing results. It develops a unitary- and PVHS-based strategy to deduce integrality and finiteness, then extends the result to semisimple and non-semisimple cases via Putman–Wieland asymptotics. The work yields arithmetic consequences, including instances of the Fontaine–Mazur conjecture and nonliftable residual representations, and provides asymptotic PW results that connect to questions in geometric topology. Together, these results offer a robust framework linking MCG dynamics, Hodge theory, and arithmetic geometry, with broad implications for isomonodromic deformations and the structure of local systems on moduli spaces.

Abstract

Let be an orientable surface of genus with punctures. We study actions of the mapping class group of via Hodge-theoretic and arithmetic techniques. We show that if is a representation whose conjugacy class has finite orbit under the mapping class group, and , then 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.
Paper Structure (61 sections, 57 theorems, 147 equations, 3 figures)

This paper contains 61 sections, 57 theorems, 147 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Results for surface groups
  4. An application to low rank local systems on $\mathscr M_{g,n}$
  5. The Putman-Wieland conjecture
  6. Arithmetic applications
  7. A consequence for free groups
  8. Cohomological results
  9. Prior work
  10. MCG-finite representations and Painlevé VI
  11. MCG-finite representations and geometric topology
  12. The Putman-Wieland conjecture
  13. Outline of the proof
  14. Step 1. The unitary case.
  15. Step 2. The semisimple case.
  16. ...and 46 more sections

Key Result

Theorem 1.2.1

For $g, n, r\geq 0$, let be a MCG-finite representation. If $r<\sqrt{g+1}$, then $\rho$ has finite image.

Figures (3)

  • Figure 1: The geography of known MCG-finite representations. \ref{['theorem:finite-image']} shows that the region $r<\sqrt{g+1}$ contains only MCG-finite representations with finite image. As explained in \ref{['question:bounds']}, there are examples of MCG-finite rank $r$ representations with infinite image for $r\gg g$. For $r$ polynomial in $g$, all known examples have virtually solvable image. For $r$ exponential in $g$, there are interesting MCG-finite irreps with infinite image.
  • Figure 2: A diagram depicting the structure of the proof of the main result, \ref{['theorem:finite-image']}, in the shape of a boat.
  • Figure 3: A schematic diagram of the mixed Hodge structure on the cohomology of a unitary local system.

Theorems & Definitions (148)

  • Definition 1.1.1
  • Theorem 1.2.1
  • Remark 1.2.2
  • Corollary 1.2.3
  • Corollary 1.3.1
  • Theorem 1.4.1
  • Theorem 1.4.2
  • Corollary 1.6.1
  • proof
  • Remark 1.6.2
  • ...and 138 more