Difference between revisions of "Effective action of finite group of diffeomorphisms has free point"
(New page: ==Statement== Suppose <math>M</math> is a connected differential manifold and <math>G</math> is a finite subgroup of the selfdiffeomorphism group of <math>M</mat...) 
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Latest revision as of 19:39, 18 May 2008
Contents
Statement
Suppose is a connected differential manifold and is a finite subgroup of the selfdiffeomorphism group of . In other words, is a finite group acting on such that no nonidentity 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:
 Fixedpoint 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.