Geodesic
Translation surfaces and their orbit closures: An introduction for a broad audience
Alex Wright1
1Stanford University, USA
EMS surveys in mathematical sciences, Volume 2 (2015) no. 1, pp. 63-108

See the original article notice from the EMS Press source

Translation surfaces can be defined in an elementary way via polygons, and arise naturally in the study of various basic dynamical systems. They can also be defined as differentials on Riemann surfaces, and have moduli spaces called strata that are related to the moduli space of Riemann surfaces. There is a GL(2,R) action on each stratum, and to solve most problems about a translation surface one must first know the closure of its orbit under this action. Furthermore, these orbit closures are of fundamental interest in their own right, and are now known to be algebraic varieties that parameterize translation surfaces with extraordinary algebro-geometric and flat properties. The study of orbit closures has greatly accelerated in recent years, with an influx of new tools and ideas coming from diverse areas of mathematics.
DOI: 10.4171/emss/9
Classification: 37-XX
Keywords: Translation surface, Teichmüller dynamics, Abelian differential, Riemann surface, moduli space

Alex Wright  1

1 Stanford University, USA 
Alex Wright. Translation surfaces and their orbit closures: An introduction for a broad audience. EMS surveys in mathematical sciences, Volume 2 (2015) no. 1, pp. 63-108. doi: 10.4171/emss/9
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