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Geometric and Functional Analysis

, Volume 18, Issue 6, pp 1988–2016 | Cite as

The Arnoux–Yoccoz Teichmüller disc

  • Pascal HubertEmail author
  • Erwan Lanneau
  • Martin Möller
Open Access
Article

Abstract.

We prove that the Teichmüller disc stabilized by the Arnoux-Yoccoz pseudo-Anosov diffeomorphism contains at least two closed Teichmüller geodesics. This proves that the corresponding flat surface does not have a cyclic Veech group.

In addition, we prove that this Teichmüller disc is dense inside the hyperelliptic locus of the connected component \({\mathcal{H}}^{\rm odd}\)(2,2) . The proof uses Ratner’s theorems.

Rephrasing our results in terms of quadratic differentials, we show that there exists a holomorphic quadratic differential, on a genus 2 surface, with the two following properties:
  1. 1.

    The Teichmüller disc is dense inside the moduli space of holomorphic quadratic differentials (which are not the global square of any Abelian differentials).

     
  2. 2.

    The stabilizer of the \({\rm PSL}_2\) (\({\mathbb{R}}\))-action contains two non-commuting pseudo-Anosov diffeomorphisms.

     

Keywords and phrases:

Abelian differentials Veech group Pseudo-Anosov diffeomorphism Teichmüller disc Ratner’s theorems 

AMS Mathematics Subject Classification:

Primary: 32G15 Secondary: 30F30, 57R30, 37D40 

Copyright information

© Birkhäuser Verlag, Basel 2009

Authors and Affiliations

  1. 1.Laboratoire d’Analyse, Topologie et Probabilités (LATP)Case cour A Faculté de Saint JérômeMarseille cedex 20France
  2. 2.Centre de Physique Théorique (CPT), UMR CNRS 6207Université du Sud Toulon-Var and Fédération de Recherches des Unités de Mathématiques de MarseilleMarseille Cedex 9France
  3. 3.Max-Planck-Institut für MathematikBonnGermany

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