Abstract
The family of translation surfaces (X g , ω 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 , ω g ) converge to a surface (X ∞, ω ∞) of infinite genus and finite area. We study the exchange on infinitely many intervals that arises from the vertical flow on (X ∞, ω ∞) and compute the affine group of (X ∞, ω ∞), which has an index 2 cyclic subgroup generated by a hyperbolic element.
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Bowman, J.P. The complete family of Arnoux–Yoccoz surfaces. Geom Dedicata 164, 113–130 (2013). https://doi.org/10.1007/s10711-012-9762-9
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DOI: https://doi.org/10.1007/s10711-012-9762-9
Keywords
- Translation surface
- Abelian differential
- Affine homeomorphism
- Veech group
- Interval exchange transformation
- Triangulation