  Given a finite subgroup G of the mapping class group of a surface S, the
Nielsen realization problem asks whether G can be realized as a finite group of
homeomorphisms of S. In 1983, Kerckhoff showed that for S a finite-type
surface, any finite subgroup G may be realized as a group of isometries of some
hyperbolic metric on 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. Finally,
we show that compact subgroups of the mapping class group of S are finite, and
locally compact subgroups are discrete.

Geometric Topology;Group Theory