Extending free actions of finite groups on surfaces

Abstract

The present work introduces new perspectives in order to extend orientation-preserving finite group actions from oriented surfaces to 3-manifolds. We consider the Schur multiplier associated to a finite group G in terms of two-dimensional principal G-bordisms, called G-cobordisms. We are interested in the question of when a free action of a finite group on a closed oriented surface extends to a non-necessarily free action on a 3-manifold. We show the answer to this question is always affirmative for abelian, dihedral, symmetric and alternating groups. As an application of our methods, we show some particular cases for non-necessarily free actions of abelian groups and dihedral groups on surfaces.