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Algebrogeometric subgroups of mapping class groups

Philippe Eyssidieux, Louis Funar

Abstract

We provide new constraints for algebrogeometric subgroups of mapping class groups, namely images of fundamental groups of curves under complex algebraic maps to the moduli space of smooth curves. Specifically, we prove that the restriction of an infinite, finite rank unitary representation of the mapping class group to an algebrogeometric subgroup should be infinite, when the genus is at least 3. In particular the restriction of most Reshetikhin-Turaev representations of the mapping class group to such subgroups is infinite. To this purpose we use deep work of Gibney, Keel and Morrison to constrain the Shafarevich morphism associated to a linear representation of the fundamental group of the compactifications of the moduli stack of smooth curves studied in our previous work. As an application we prove that universal covers of most of these compactifications are Stein manifolds.

Algebrogeometric subgroups of mapping class groups

Abstract

We provide new constraints for algebrogeometric subgroups of mapping class groups, namely images of fundamental groups of curves under complex algebraic maps to the moduli space of smooth curves. Specifically, we prove that the restriction of an infinite, finite rank unitary representation of the mapping class group to an algebrogeometric subgroup should be infinite, when the genus is at least 3. In particular the restriction of most Reshetikhin-Turaev representations of the mapping class group to such subgroups is infinite. To this purpose we use deep work of Gibney, Keel and Morrison to constrain the Shafarevich morphism associated to a linear representation of the fundamental group of the compactifications of the moduli stack of smooth curves studied in our previous work. As an application we prove that universal covers of most of these compactifications are Stein manifolds.

Paper Structure

This paper contains 28 sections, 36 theorems, 40 equations.

Table of Contents

  1. Introduction and statements
  2. Motivation
  3. Definitions
  4. Restricting unitary representations
  5. Restricting Reshetikhin-Turaev representations
  6. The Stein property
  7. Teichmüller curves and Veech groups
  8. Degree two cohomology classes
  9. Proof of Theorem \ref{['unitary']}
  10. Stacky compactifications of moduli space of smooth curves
  11. Shafarevich morphisms for stacks
  12. Application to ${\overline{\mathcal{M}_{g}^{n}}}[\mathbf k]$
  13. Proof of Theorem \ref{['corointro1']}
  14. Proof of Theorem \ref{['theointro1']}
  15. Idea of the proof and evidence from the F-Conjecture
  16. ...and 13 more sections

Key Result

Theorem 1

Let $g\geq 3$, $\rho$ be a finite rank unitary representation of ${\rm{PMod}}(\Sigma_{g}^n)$ with infinite image. Assume furthermore that the restriction of $\rho$ to every geometric surface subgroup is also infinite, when $n\geq 1$. Then for every non-isotrivial algebraic family $f:C \to \mathcal{M

Theorems & Definitions (71)

  • Theorem 1
  • Remark 1.1
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Theorem 5
  • Corollary 6
  • Corollary 7
  • Theorem 8
  • Remark 1.2
  • ...and 61 more