Nielsen realization problem

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The Nielsen realization problem is a 1932 conjecture by Jakob Nielsen, who posed it as a question.

Given a surface, we can divide the homeomorphisms of the surface to itself in isotopy classes. The conjecture asks whether a finite group of these classes of a surface can be realized as the isometry group of a hyperbolic surface. This was shown to be true by Steven Kerckhoff in 1983.

 

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