Teichmüller discs with completely degenerate Kontsevich–Zorich spectrum

  • David Aulicino

    University of Chicago, USA

Abstract

We reduce a question of Eskin–Kontsevich–Zorich and Forni–Matheus–Zorich, which asks for a classification of all -invariant ergodic probability measures with completely degenerate Kontsevich–Zorich spectrum, to a conjecture of Möller's. Let be the subset of the moduli space of Abelian differentials whose elements have period matrix derivative of rank one. There is an -invariant ergodic probability measure with completely degenerate Kontsevich–Zorich spectrum, i.e. , if and only if has support contained in . We approach this problem by studying Teichmüller discs contained in . We show that if generates a Teichmüller disc in , then is completely periodic. Furthermore, we show that there are no Teichmüller discs in , for , and the two known examples of Teichmüller discs in , for , are the only two such discs in those genera. Finally, we prove that if there are no genus five Veech surfaces generating Teichmüller discs in , then there are no Teichmüller discs in , for .

Cite this article

David Aulicino, Teichmüller discs with completely degenerate Kontsevich–Zorich spectrum. Comment. Math. Helv. 90 (2015), no. 3, pp. 573–643

DOI 10.4171/CMH/365