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Rotational component spaces for infinite-type translation surfaces

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Abstract

Finite translation surfaces can be classified by the order of their singularities. When generalizing to infinite translation surfaces, however, the notion of order of a singularity is no longer well-defined and has to be replaced by new concepts. This article discusses the nature of two such concepts, recently introduced by Bowman and Valdez: linear approaches and rotational components. We show that there is a large flexibility in the spaces of rotational components and even more in the spaces of linear approaches. In particular, we prove that every finite topological space arises as space of rotational components. However, this space will still not contain enough information to describe an infinite translation surface. We showcase this through an uncountable family with the same space of rotational components but different spaces of linear approaches. Additionally, we study several known and new examples to illustrate the concept of linear approaches and rotational components.

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Acknowledgements

The idea for this project was developed during the semester program Low-dimensional Topology, Geometry, and Dynamics at the Institute for Computational and Experimental Research in Mathematics in Providence. Most of this work was done while the first and the third author were visiting the Karlsruhe Institute of Technology, funded by a Visiting Researcher Scholarship of the Karlsruhe House of Young Scientists. The second author was partially supported within the project “Dynamik unendlicher Translationsflächen”in the “Juniorprofessoren-Programm”of the Ministry of Science, Research and the Arts of Baden-Wuerttemberg.

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Authors and Affiliations

  1. Department of Mathematics, Cornell University, Malott Hall, Ithaca, NY, 14853, USA

    Lucien Clavier & Chenxi Wu

  2. Institut für Algebra und Geometrie, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), Kaiserstraße 12, 76131, Karlsruhe, Germany

    Anja Randecker

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  1. Lucien Clavier
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  2. Anja Randecker
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  3. Chenxi Wu
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Correspondence to Anja Randecker.

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Clavier, L., Randecker, A. & Wu, C. Rotational component spaces for infinite-type translation surfaces. Geom Dedicata 201, 57–80 (2019). https://doi.org/10.1007/s10711-018-0381-y

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