On the geometry and arithmetic of infinite translation surfaces

Journal of Singularities
volume 9 (2014), 226-244

Received 29 January 2012. Received in revised form 14 October 2013.

DOI: 10.5427/jsing.2014.9r

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Abstract:

Precompact translation surfaces, i.e., closed surfaces which carry a translation atlas outside of finitely many finite angle cone points, have been intensively studied for about 25 years now. About 5 years ago the attention was also drawn to general translation surfaces. In this case the underlying surface can have infinite genus, the number of finite angle cone points of the translation structure can be infinite, and there can be singularities which are not finite angle cone points. There are only a few invariants one classically associates with precompact translation surfaces, among them certain number fields, i.e., fields which are finite extensions of the rational numbers. These fields are closely related to each other; they are often even equal. We prove by constructing explicit examples that most of the classical results for the fields associated with precompact translation surfaces fail in the realm of general translation surfaces and investigate the relations among these fields. A very special class of translation surfaces are so called square-tiled surfaces or origamis. We give a characterisation for infinite origamis.

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