The complete family of Arnoux-Yoccoz surfaces

Abstract

The family of translation surfaces $(X_g,\omega_g)$ constructed by Arnoux and Yoccoz from self-similar interval exchange maps encompasses one example from each genus $g$ greater than or equal to $3$. We triangulate these surfaces and deduce general properties they share. The surfaces $(X_g,\omega_g)$ converge to a surface $(X_\infty,\omega_\infty)$ of infinite genus and finite area. We study the exchange on infinitely many intervals that arises from the vertical flow on $(X_\infty,\omega_\infty)$ and compute the affine group of $(X_\infty,\omega_\infty)$, which has an index $2$ cyclic subgroup generated by a hyperbolic element.