Nielsen realization problem
Encyclopedia
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 .
s 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.
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. pointed out a gap in the argument, and in 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.
Statement
Given an oriented surface, we can divide group Diff(S) the diffeomorphismDiffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
s of the surface to itself in isotopy
Homotopy
In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...
classes to get the mapping class group
Mapping class group
In mathematics, in the sub-field of geometric topology, the mapping class groupis an important algebraic invariant of a topological space. Briefly, the mapping class group is a discrete group of 'symmetries' of the space.-Motivation:...
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
Teichmüller space
In mathematics, the Teichmüller space TX of a topological surface X, is a space that parameterizes complex structures on X up to the action of homeomorphisms that are isotopic to the identity homeomorphism...
. 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 Teichmüller space
Teichmüller space
In mathematics, the Teichmüller space TX of a topological surface X, is a space that parameterizes complex structures on X up to the action of homeomorphisms that are isotopic to the identity homeomorphism...
(with the Teichmüller metric) is negatively curved. pointed out a gap in the argument, and in 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.