Effective action of finite group of diffeomorphisms has free point

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Statement

Suppose M is a connected differential manifold and G is a finite subgroup of the self-diffeomorphism group of M. In other words, G is a finite group acting on M such that no non-identity element of G acts trivially on M.

Then, the action of G on M has a free point: a point whose isotropy subgroup (stabilizer) is trivial.

Definitions used

Fill this in later

Facts used

We use two main facts:

  1. Fixed-point set of finite group of diffeomorphisms is closed submanifold
  2. A finite union of submanifolds of codimension at least one is proper.

Proof

For every g \in G other than the identity element, consider the set M^g of elements of M fixed by the cyclic subgroup generated by g. By fact (1) above, this is a closed submanifold, and by the effectiveness assumption, it cannot be the whole of M. Since M is connected, it cannot be of full dimension, otherwise it would be both closed and open. Hence M^g is either empty, or is a submanifold of codimension at least 1.

Now, consider:

\bigcup_{g \in G \setminus \{ e \}} M^g

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 g \in G, and this is the desired free point.