Effective action of finite group of diffeomorphisms has free point
Suppose is a connected differential manifold and is a finite subgroup of the self-diffeomorphism group of . In other words, is a finite group acting on such that no non-identity element of acts trivially on .
Then, the action of on has a free point: a point whose isotropy subgroup (stabilizer) is trivial.
Fill this in later
We use two main facts:
- Fixed-point set of finite group of diffeomorphisms is closed submanifold
- A finite union of submanifolds of codimension at least one is proper.
For every other than the identity element, consider the set of elements of fixed by the cyclic subgroup generated by . By fact (1) above, this is a closed submanifold, and by the effectiveness assumption, it cannot be the whole of . Since is connected, it cannot be of full dimension, otherwise it would be both closed and open. Hence is either empty, or is a submanifold of codimension at least .
This is a finite union of submanifolds of codimension at least 1, so is a proper subset. Thus, we can pick a point not fixed by any , and this is the desired free point.