Nielsen realization problem

Template:Short description The Nielsen realization problem is a question asked by Template:Harvs about whether finite subgroups of mapping class groups can act on surfaces, that was answered positively by Template:Harvs.

Statement

Given an oriented surface, we can divide the group Diff(S), the group of diffeomorphisms of the surface to itself, into isotopy classes to get the mapping class group π₀(Diff(S)). The conjecture asks whether a finite subgroup 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.

History

Template:Harvs asked whether finite subgroups of mapping class groups can act on surfaces. Template:Harvtxt claimed to solve the Nielsen realization problem but his proof depended on trying to show that Teichmüller space (with the Teichmüller metric) is negatively curved. Template:Harvtxt pointed out a gap in the argument, and Template:Harvtxt showed that Teichmüller space is not negatively curved. Template:Harvs gave a correct proof that finite subgroups of mapping class groups can act on surfaces using left earthquakes.