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The Arnoux-Yoccoz Teichmüller disc

Erwan Lanneau, Pascal Hubert, Martin Möller

Number 2
Author Martin Möller
Year 2007

We prove that the Teichmüller disc stabilized by the Arnoux-Yoccoz pseudo-Anosov possesses two transverse hyperbolic directions. This proves that the corresponding flat surface has not a cyclic Veech group. In addition, we prove that this Teichmüller disc is dense inside the hyperelliptic locus of the connected component Hodd(2; 2).

We rephrase 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 PSL2(R)-action contains two non commuting pseudo-Anosov diffeomorphisms.

The proof uses Ratner's theorems.

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