Mathematics Colloquia and Seminars

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Complex projective structure is structure on a closed orientable surface, which is a generalization of hyperbolic structure. Every discrete and faithful representation from the surface group to PSL(2,R) corresponds to a unique complete hyperbolic structure. On the other hand, every representation from the surface group to PSL(2,C) with some much milder conditions corresponds to infinitely many complex projective structures. In the case that the representation is an isomorphism onto a quasifuchsian group, its corresponding complex projective structures are beautifully characterized by William Goldman, and his key argument uses bundles and Euler classes. I will give another proof of the characterization theorem using only hyperbolic geometry techniques.