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Conformal Geometry and Dynamics

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Immersions and translation structures I: The space of structures on the pointed disk


Author: W. Patrick Hooper
Journal: Conform. Geom. Dyn. 22 (2018), 235-270
MSC (2010): Primary 57M50; Secondary 30F30, 32G15, 37E99, 06B23
DOI: https://doi.org/10.1090/ecgd/326
Published electronically: October 23, 2018
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Abstract: We define a moduli space of translation structures on the open topological disk with a basepoint and endow it with a locally-compact metrizable topology. We call this the immersive topology, because it is defined using the concept of immersions: continuous maps between subsets of translation surfaces that respect the basepoints and the translation structures. Immersions induce a partial ordering on the moduli space, and we prove the ordering is nearly a complete lattice in the sense of order theory; the space is only missing a minimal element. Subsequent articles will uncover more structure and develop a topology on the space of all translation structures.


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Additional Information

W. Patrick Hooper
Affiliation: Department of Mathematics, The City College of New York, New York, New York, 10031
Email: whooper@ccny.cuny.edu

DOI: https://doi.org/10.1090/ecgd/326
Received by editor(s): May 29, 2014
Received by editor(s) in revised form: June 2, 2015, May 27, 2016, and August 7, 2018
Published electronically: October 23, 2018
Additional Notes: Support was provided by N.S.F. Grants DMS-1101233 and DMS-1500965 as well as a PSC-CUNY Award (funded by The Professional Staff Congress and The City University of New York).
Article copyright: © Copyright 2018 W. Patrick Hooper

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