Definition. A group splits freely if acts on a tree without global fixed point and such that every edge stailizer is trivial. If does not split freely, then is called freely indecomposable.
Example. . Equivalently, acts on without global fixed points. So splits freely.
If but splits freely, then for .
Definition. The rank of is the minimal such that surjects .
It is clear that .
Grushko’s Lemma. Suppose is surjective and is minimal. If , then such that for .
Pf. Let be simplicial and let be a graph of spaces with vertex spaces and edge space a point. So where .
Let be a graph so that and realize as a simplicial map . Let . Because is minimal, is a forest, contained in . The goal is to modify by a homotopy to reduce the number of connected components of .
Let be the component that contains . Let be some other component. Let a path in from to .
Look at . Because is surjective, there exists such that . Therefore if , then is null-homotopic in and gives a path from to .
We can write as a concaternation as such that for each , . By the Normal Form Theorem, there exists such that is null-homotopic in .
We can now modify by a homotopy so that . Therefore and the number of components of has gone down. By induction, we can choose so that is a tree. Now factors through . Then and there is a unique vertex of that maps to . So every simple loop in is either contained in or as required.
An immediate consequence is that .
Grushko’s Theorem. Let be finitely generated. Then where each is freely indecomposable and is free. Furthermore, the integers and are unique and the are unique up to conjugation and reordering.
Pf. Existence is an immediate corollary of the fact that rank is additive.
Suppose . Let be the graph of groups. Let be the Bass-Serre tree of .
Consider the action of on . Because is freely indecomposable, stabilize a vertex of . Therefore is conjugate into some .
Now consider the action of on . is a graph of groups with underlying graph , say, and is a free factor in . But there is a covering map that induces a surjection . Therefore, . The other inequality can be obtained by switching and .