The Nielsen realization problem

Abstract

Closed, oriented surfaces of genus g > 2 admit many hyperbolic (constant Gaussian curvature -1) metrics in contrast to Mostow's rigidity theorems in higher dimensions. Only special hyperbolic surfaces have non-trivial groups of isometries, but many different, non-isomorphic groups arise for different symmetric metrics. The group of isometries of a closed hyperbolic surface M2 is always finite and the only isometry isotopic to the identity is the identity itself. As a result, hyperbolic surfaces with non-trivial groups of isometries have been a primary source for the construction of finite subgroups of the group of isotopy classes of diffeomorphisms of M2, ?TDiff(M2). An old question, usually referred to as the Nielsen Realization Problem, is whether every such finite subgroup arises as a group of isometries of some hyperbolic surface. In this paper we answer the question in the affirmative.