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

Hooper, W.

doi:10.1090/ecgd/326

Immersions and translation structures I: The space of structures on the pointed disk
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by W. Patrick Hooper

Conform. Geom. Dyn. 22 (2018), 235-270

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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