  We obtain an effective enumeration of the family of finitely generated groups
admitting a faithful, properly discontinuous action on some 2-manifold
contained in the sphere. This is achieved by introducing a type of group
presentation capturing exactly these groups.
  Extending this in a companion paper, we find group presentations capturing
the planar finitely generated Cayley graphs. Thus we obtain an effective
enumeration of these Cayley graphs, yielding in particular an affirmative
answer to a question of Droms et al.
