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Veech Groups and Translation Coverings

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ISBN: 9783731501800 Year: Pages: X, 136 p. DOI: 10.5445/KSP/1000038927 Language: ENGLISH
Publisher: KIT Scientific Publishing
Subject: Mathematics
Added to DOAB on : 2019-07-30 20:01:59
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Abstract

A translation surface is obtained by taking plane polygons and gluing their edges by translations. We ask which subgroups of the Veech group of a primitive translation surface can be realised via a translation covering. For many primitive surfaces we prove that partition stabilising congruence subgroups are the Veech group of a covering surface. We also address the coverings via their monodromy groups and present examples of cyclic coverings in short orbits, i.e. with large Veech groups.

Invariants of complex and p-adic origami-curves

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ISBN: 9783866444829 Year: Pages: VI, 74 p. DOI: 10.5445/KSP/1000015949 Language: ENGLISH
Publisher: KIT Scientific Publishing
Subject: Mathematics
Added to DOAB on : 2019-07-30 20:01:59
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Origamis (also known as square-tiled surfaces) are Riemann surfaces which are constructed by glueing together finitely many unit squares. By varying the complex structure of these squares one obtains easily accessible examples of Teichmüller curves in the moduli space of Riemann surfaces.Different Teichmüller curves can be distinguished by several invariants, which are explicitly computed. The results are then compared to a p-adic analogue where Riemann surfaces are replaced by Mumford curves.

Geometry and topology of wild translation surfaces

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ISBN: 9783731504566 Year: Pages: 151 p. DOI: 10.5445/KSP/1000050964 Language: ENGLISH
Publisher: KIT Scientific Publishing
Subject: Mathematics
Added to DOAB on : 2019-07-30 20:02:02
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Abstract

A translation surface is a two-dimensional manifold, equipped with a translation structure. It can be obtained by considering Euclidean polygons and identifying their edges via translations. The vertices of the polygons form singularities if the translation structure can not be extended to them. We study translation surfaces with wild singularities, regarding the topology (genus and space of ends), the geometry (behavior of the singularities), and how the topology and the geometry are related.

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