# Effective action of finite group of diffeomorphisms has free point

## Contents

## Statement

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.

## Definitions used

*Fill this in later*

## Facts used

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.

## Proof

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 .

Now, consider:

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.