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

Statement

Given an oriented surface, we can divide the group Diff(''S''), the group of

diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two ...

s of the surface to itself, into isotopy classes to get the

mapping class group In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space. Mo ...

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 In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S, is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmüll ...

. An equivalent way of stating the question asks whether every finite subgroup of the mapping class group fixes some point of Teichmüller space.

History

asked whether finite subgroups of mapping class groups can act on surfaces. claimed to solve the Nielsen realization problem but his proof depended on trying to show that

(with the Teichmüller metric) is negatively curved. pointed out a gap in the argument, and showed that Teichmüller space is not negatively curved. gave a correct proof that finite subgroups of mapping class groups can act on surfaces using left earthquakes.

References

* * * * * * {{DEFAULTSORT:Nielsen Realization Problem Geometric topology Homeomorphisms Theorems in topology