The Nielsen Realization Problem

Abstract

If M is a closed, oriented 2-manifold of genus g> 2, then it admits many hyperbolic metrics (metrics of constant curvature- 1). In special cases such a metric possesses a nontrivial group of symmetries, of isometries to itself. The group of isometries of a closed hyperbolic manifold is always finite and the only isometry isotopic to the identity is the identity itself. Thus a group of symmetries of a hyperbolic surface determines an isomorphic finite subgroup of the group of isotopy classes of diffeomorphisms of M. The purpose of this paper is to announce a positive solution to the Nielsen Realization Problem that the converse is true, i.e.; the THEOREM 1. Every finite subgroup of 7r0Diff Mg can be realized as a group of isometries of some hyperbolic structure on Mg. For g> 2 the map Diff Mg — • 7r0 Diff Mg is a homotopy equivalence, but it is unknown whether or not 7r0 Diff Mg can be lifted back into Diff Mg as a subgroup. Theorem 1 solves the lifting problem for finite subgroups of n0 Diff M. We will call 7r0Diff Mg the modular groupe (Mod) since it is the natural