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From Period Coordinates to Teichmueller Distance

Ian Frankel, University of Chicago
Tuesday, December 5, 2017 - 1:30pm to 3:30pm
PDL C-401

A half-translation surface is a closed oriented surface formed by gluing together polygons in the plane by identifying parallel edges. Such a surface carries a Riemann surface structure and a flat metric with singularities - cone points at the vertices of the polygons. They arise naturally in the study of the moduli space of Riemann surfaces, particularly in the study of the Teichmuller metric (which measures how far from conformal equivalence two surfaces are) and its geodesic flow. We define a metric on the moduli spaces of half-translation surfaces in terms of the Euclidean geometry of polygons, and show that their projections to the moduli spaces of Riemann surfaces, with Teichm\"uller's metric, are locally Holder.