Suppose a group is acting properly and cocompactly on a metric space X, by isometries.
(Understand: proper, cocompact, isometric action)
There are only finitely many conjugacy classes of the isotropy subgroups in
Since the action is cocompact, there is a compact set K whose translates cover X.
For each , choose small (enough) balls .
How small do we want the balls?
Small enough, such that all but finitely many of the translates of the ball is disjoint from the ball.
In the picture, the green blob around x in K is an example of . It is small enough such that is disjoint from . The ball is small enough such that most almost this should happen (except finitely many of them).
How do we know that there is such a ball? This is by definition of proper action. Since the action is proper, by definition, we will get such balls for each x in X.
Intuitively a proper action is proper, that is, most of them (the isometries) move balls considerably.
Examples to keep in mind: Integer translation on the real line is proper. Rational translation on the real line is not proper.
is an open cover of K. Hence it has a finite subcover (as K is compact).
Choose finitely many balls , such that they cover K.
These finitely many balls have two properties:
- They cover K
- for finitely many
Thus each member of has this property : hit all of the balls in the previously found cover by it. At least one of them will not move a lot (that is the translate will have a non-empty intersection with the ball)
Also notice that is finite (as there are finitely many balls and for each ball there are finitely many .
Suppose be any point. There is a such that .
Then . This is the element of K that goes to x under
Now consider the isotropy subgroup , that is group of all isometries that fixes x.
Notice that is the isotropy subgroup of . Why? Suppose . We will show that .
We can similarly show the converse.
Hence we have shown that if x is pulled back into K by some conjugating the isotropy subgroup of x by that gives the isotropy subgroup of the pulled back element.
But this is the subgroup that does not move the ball in the cover containing in a disjoint manner. Hence it must be inside , and hence finite as was finite.