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Hodge and Teichmüller

Jeremy Kahn, Alex Wright

Abstract

We consider the derivative $Dπ$ of the projection $π$ from a stratum of Abelian or quadratic differentials to Teichmüller space. A closed one-form $η$ determines a relative cohomology class $[η]_Σ$, which is a tangent vector to the stratum. We give an integral formula for the pairing of of $Dπ([η]_Σ)$ with a cotangent vector to Teichmüller space (a quadratic differential). We derive from this a comparison between Hodge and Teichmüller norms, which has been used in the work of Arana-Herrera on effective dynamics of mapping class groups, and which may clarify the relationship between dynamical and geometric hyperbolicity results in Teichmüller theory.

Hodge and Teichmüller

Abstract

We consider the derivative of the projection from a stratum of Abelian or quadratic differentials to Teichmüller space. A closed one-form determines a relative cohomology class , which is a tangent vector to the stratum. We give an integral formula for the pairing of of with a cotangent vector to Teichmüller space (a quadratic differential). We derive from this a comparison between Hodge and Teichmüller norms, which has been used in the work of Arana-Herrera on effective dynamics of mapping class groups, and which may clarify the relationship between dynamical and geometric hyperbolicity results in Teichmüller theory.

Paper Structure

This paper contains 5 sections, 10 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. The derivative formula
  3. The principal stratum
  4. Arbitrary strata of quadratic differentials
  5. Norm comparisons.

Key Result

Theorem 1.1

Let $\eta=\eta^{1,0}+\eta^{0,1}$ be closed as above. Let $\{\gamma_1, \ldots, \gamma_s\}$ denote small disjoint positively oriented loops around the zeros of $\omega$, and let $X'$ be the complement in $X$ of the discs that they bound. Then the pairing of $(D\pi)[\eta]_\Sigma$ with a quadratic diffe where $F_j(z) = \int_{z_j}^z \eta$ is defined by integrating along paths in the disc containing $z_

Theorems & Definitions (21)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Theorem 1.4
  • Proposition 1.5
  • proof
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 11 more