Neitzke, Andrew2016-10-132018-01-222016-10-132018-01-222016-05May 2016http://hdl.handle.net/2152/41629The 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.application/pdfenGeometric structuresAbelianizationThesis2016-10-13