Statement
Given an oriented surface, we can divide group Diff(S) the diffeomorphisms of the surface to itself in isotopy classes to get the mapping class group π0(Diff(S)). The conjecture asks whether a finite group of the mapping class group of a surface can be realized as the isometry group of a hyperbolic metric on the surface.
The mapping class group acts on Teichmüller space. An equivalent way of stating the question asks whether every finite subgroup of the mapping class group fixes some point of Teichmüller space.
Read more about this topic: Nielsen Realization Problem
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