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
Famous quotes containing the word statement:
“One is apt to be discouraged by the frequency with which Mr. Hardy has persuaded himself that a macabre subject is a poem in itself; that, if there be enough of death and the tomb in one’s theme, it needs no translation into art, the bold statement of it being sufficient.”
—Rebecca West (1892–1983)
“The new statement is always hated by the old, and, to those dwelling in the old, comes like an abyss of skepticism.”
—Ralph Waldo Emerson (1803–1882)
“If we do take statements to be the primary bearers of truth, there seems to be a very simple answer to the question, what is it for them to be true: for a statement to be true is for things to be as they are stated to be.”
—J.L. (John Langshaw)