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Title:
A bumpy ride through the space of half-translation structures
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Abstract:
As you deform a half-translation surface with non-compact ends, its combinatorial features jump whenever a vertical saddle connection appears. The associated shear coordinates on the Teichmüller space of the surface jump too, giving the algebra of shear coordinates its celebrated cluster structure. When you deform a compact half-translation surface, you get an even bumpier ride, with jumps at a dense set of times. I'll set out some ideas for keeping track of these jumps, and show hints of a generalized cluster structure on the algebra of shear coordinates.
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