I am new to orbifolds. By reading the definition (classical ones, not in terms of stacks or groupoids), I am wondering why only finite group action is allowed in the definition of local charts. I am expecting that a quotient manifold is an orbifold where the action is proper. However, it seems that it is not the case unless the action is, in fact, properly discontinuous. It seems that the crux here is that stabilizers cannot be too big and need to be finite.
My questions are:
- is the finite group action absolutely necessary, or is it merely a technical convenience?
- What happens if compact groups are allowed in local charts? Is there already such generalization?
Thanks.
Best Answer
One purpose of the definition of an orbifold was to be able to describe quotient objects of general properly discontinuous group actions, such as triangle group actions on the 2-sphere, on the euclidean plane, or on the hyperbolic plane.
In the special case where the group action is also free, a very nice theory already existed, namely the theory of regular covering maps of manifolds. In that situation, every point stabilizer is trivial.
In the general case where the group action may not be free, proper discontinuity implies that the point stabilizers are finite. Those point stabilizers of the group action become the finite groups that label points in the singular locus of the quotient orbifold. For a properly discontinuous group action on a surface, for example, its easy to prove that the groups that can arise as point stabilizers are finite cyclic groups and finite dihedral groups. For that reason, the groups that label the singular locus of an orbifold of dimension 2 are all either finite cyclic groups or finite dihedral groups.