The Arnoux-Yoccoz Teichmueller disc

Abstract

We prove that the Teichmueller disc stabilized by the Arnoux-Yoccoz pseudo-Anosov diffeomorphism contains at least two closed Teichmueller geodesics. This proves that the corresponding flat surface does not have a cyclic Veech group. In addition, we prove that this Teichmueller disc is dense inside the hyperelliptic locus of the connected component H^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 Teichmueller 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 PSL(2,R)-action contains two non-commuting pseudo-Anosov diffeomorphisms.