Immersions and the space of all translation structures

W. Patrick Hooper

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arXiv · 1310.5193

Immersions and the space of all translation structures

W. Patrick Hooper

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Abstract

A translation structure on a surface is an atlas of charts to the plane so that the transition functions are translations. We allow our surfaces to be non-compact and infinite genus. We endow the space of all pointed surfaces equipped with a translation structure with a topology, which we call the immersive topology because it is related to the manner in which disks can be immersed into such a surface. We prove that a number of operations typically done to translation surfaces are continuous with respect to the topology. We show that the topology is Hausdorff, and that the collection of surfaces with a fixed lower bound on the injectivity radius at the basepoint is compact.

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W. Patrick Hooper. 2013-10-19. Immersions and the space of all translation structures. https://arxiv.org/abs/1310.5193

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