2006/11/21 by Pascal Hubert, Hubert, Pascal, Erwan Lanneau +4
Mathematics · #32G15 (Primary) 30F30 #37D40 (Secondary) #57R30 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Mathematical Dynamics and Fractals #Mathematics and Applications #math.DS #math.GT #msc:30F30 #msc:32G15 #msc:37D40 #msc:57R30
paper · pdf · doi:10.48550/arxiv.math/0611655
26 pages, 6 figures, to appear in GAFA. Some minor corrections made
openalex publication_date 2006/11/21 · arxiv created 2008/05/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Hodd(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.