Groups can be very complicated. One way to understand complicated objects is to break them into simpler pieces. For example, to understand large numbers, we often factorize them into their prime constituents.
How do we ‘factorize’ groups?
Instead of looking at the group, we try to examine how it ‘acts’ on some space. Imagine the group elements as ‘strikers’ and points in the space as ‘balls’. When the group elements (strikers) acts on (hits) the points in the space (balls), the balls may move a bit. We examine the movement of the balls to factorize the group!
Let’s work with a specific type of space: two dots connected with a segment.
Suppose a group G is acting on this space T.
(We are speaking very loosely here. We have not defined what is meant by group action. For the moment, the striker-ball analogy should suffice).
A chunk of the group G may ‘fix’ portion of this interval. Let us call this chunk H. H will be treated as a factor of the group G.
Next we will look at the endpoints of this interval. Some chunks larger than H (and containing H) could fix the endpoints. These will give further factorization of the group.
Notice that we are using the words ‘could’ and ‘may’. This is deliberate. Whether or not this will happen, will depend on the group under examination.
This is in nutshell what group action on trees is all about. It involves beautiful ideas from graph theory, topology and geometric group theory.