Nielsen realization for infinite-type surfaces

Abstract

Given a finite subgroup G G of the mapping class group of a surface S S , the Nielsen realization problem asks whether G G can be realized as a finite group of homeomorphisms of S S . In 1983, Kerckhoff showed that for S S a finite-type surface, any finite subgroup G G may be realized as a group of isometries of some hyperbolic metric on S S . We extend Kerckhoff’s result to orientable, infinite-type surfaces. As applications, we classify torsion elements in the mapping class group of the plane minus a Cantor set, and also show that topological groups containing sequences of torsion elements limiting to the identity do not embed continuously into the mapping class group of S S . Finally, we show that compact subgroups of the mapping class group of S S are finite, and locally compact subgroups are discrete.