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.
The Teichmüller disc is dense inside the moduli space of holomorphic quadratic differentials (which are not the global square of any Abelian differentials).
-
2.
The stabilizer of the \({\rm PSL}_2\) (\({\mathbb{R}}\))-action contains two non-commuting pseudo-Anosov diffeomorphisms.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Received: June 2007, Revision: April 2008, Accepted: April 2008
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Hubert, P., Lanneau, E. & Möller, M. The Arnoux–Yoccoz Teichmüller disc. GAFA Geom. funct. anal. 18, 1988–2016 (2009). https://doi.org/10.1007/s00039-009-0706-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00039-009-0706-y