**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 .

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