Last time we proved that if is a graph of groups, then acts on a tree , namely the underlying graph of the universal cover of . Lemmas 16, 17, and 18 describe the vertex and edge sets of and the action of . The following is immediate.
Theorem 14 (Serre): Every graph of groups is developable.
Corollary: If is a graph of groups and , then the map is injective.
The tree is called the Bass-Serre Tree of . We will usually equip with a length metric in which each edge has length 1. The approach we’ve taken is due to Scott-Wall. Here’s a sample application.
Lemma 19: Let . If is torsion, then is conjugate into or .
Proof. Let be the Bass-Serre Tree. Fix a vertex amd consider . This is a finite set.
Exercise 19: There is an that minimizes (when is torsion).
Such an is fixed by . Therefore, stabilizes a vertex. The stabilizers of the vertices are precisely the conjugates of and .
Our next goal is to understand the elements of — which are trivial and which not? We’ll start with a presentation. If is a tree, then can be thought of as a sequence of amalgamated free products. If is a rose, then can be thought of as a sequence of HNN-extensions. In general, fix a maximal tree . This determines a way of decomposing as a sequence of amalgamated products followed by a sequence of HNN-extensions. This process is sometimes called “Excision.” It follows that has a “presentation” like this:
We can write any in the following form:
where , is a loop in , and where is the terminus of and the origin of . A priori, we just know that . Suppose, say, that doesn’t originate from . Then insert stable letters corresponding to edges in that form a path from to the origin of . Now repeat.