Hodge and Teichmüller

Jeremy Kahn; Alex Wright

doi:10.3934/jmd.2022007

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2022, Volume 18: 149-160. Doi: 10.3934/jmd.2022007

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Hodge and Teichmüller

Jeremy Kahn^1, and
Alex Wright^2,

1.
Department of Mathematics, Brown University, 151, Thayer Street, Providence, RI 02912, USA
2.
Department of Mathematics, University of Michigan, 530, Church Street, Ann Arbor, MI 48109, USA

Received: July 16, 2021

Revised: November 22, 2021

Published: April 2022

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Abstract

We consider the derivative $ D\pi $ of the projection $ \pi $ from a stratum of Abelian or quadratic differentials to Teichmüller space. A closed one-form $ \eta $ determines a relative cohomology class $ [\eta]_\Sigma $, which is a tangent vector to the stratum. We give an integral formula for the pairing of $ D\pi([\eta]_\Sigma) $ 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.

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Mathematics Subject Classification: Primary: 32G15; Secondary: 30F60, 37D40.

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