Warping geometric structures and abelianizing SL(2,R) local systems
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The abelianization process of Gaiotto, Hollands, Moore, and Neitzke parameterizes SL(K,C) local systems on a punctured surface by turning them into C^\times local systems, which have a much simpler moduli space. When applied to an SL(2,R) local system describing a hyperbolic structure, abelianization produces an R^\times local system whose holonomies encode the shear parameters of the hyperbolic structure. This dissertation extends abelianization to SL(2,R) local systems on a compact surface, using tools from dynamics to overcome the technical challenges that arise in the compact setting. Thurston's shear parameterization of hyperbolic structures, which has its own technical subtleties on a compact surface, once again emerges as a special case.