Last time we proved that if \mathcal{G} is a graph of groups, then G = \pi_1 \mathcal{G} acts on a tree T, namely the underlying graph of the universal cover of X_\mathcal{G}.  Lemmas 16, 17, and 18 describe the vertex and edge sets of T and the action of G.  The following is immediate.

Theorem 14 (Serre): Every graph of groups is developable.

Corollary: If \mathcal{G} is a graph of groups and v \in V(\Gamma), then the map G_v \rightarrow G = \pi_1 \mathcal{G} is injective.

The tree T is called the Bass-Serre Tree of \mathcal{G}.  We will usually equip T 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 G = A \ast_C B.  If g \in G is torsion, then g is conjugate into A or B.

Proof. Let T be the Bass-Serre Tree.  Fix a vertex v \in T amd consider \{g^n v \mid n \in \mathbb{Z} \}.  This is a finite set.

Exercise 19: There is an x \in T that minimizes \mathrm{max}_{n \in \mathbb{Z}} d(x, g^n v) (when g is torsion).

Such an x is fixed by g.  Therefore, g stabilizes a vertex.  The stabilizers of the vertices are precisely the conjugates of A and B\Box

Our next goal is to understand the elements of G = \pi_1 \mathcal{G} — which are trivial and which not?  We’ll start with a presentation.  If \Gamma is a tree, then G can be thought of as a sequence of amalgamated free products.  If \Gamma is a rose, then G can be thought of as a sequence of HNN-extensions.  In general, fix a maximal tree \Lambda \subseteq \Gamma.  This determines a way of decomposing G as a sequence of amalgamated products followed by a sequence of HNN-extensions.  This process is sometimes called “Excision.”  It follows that G = \pi_1 \mathcal{G} has a “presentation” like this:

G = \langle \{ G_v \mid v \in V(\Gamma) \}, \{t_e \mid e \in E(\Gamma)\} \mid \\ \{t_e \partial_+^e(g)t^{-1}_e = \partial_-^e(g) \mid e \overline{t}(\Gamma), g \in G_e\}, \{t_e = 1 \mid e \in E(\Lambda)\} \rangle

We can write any g \in G in the following form:

g = g_0 t^{\epsilon_1}_{e_1} g_1 t^{\epsilon_2}_{e_2} \cdots t^{\epsilon_{n-1}}_{e_{n-1}} g_{n-1}t^{\epsilon_n}_{e_n} g_n

where \epsilon_i = \pm 1, e_1^{\epsilon_1} \cdots e_n^{\epsilon_n} is a loop in \Gamma, and g_i \in G_{v_i} where v_i is the terminus of e_i^{\epsilon_i} and the origin of e_{i+1}^{\epsilon_{i+1}}.  A priori, we just know that g = g_0 t^{\epsilon_1}_{e_1} g_1 \cdots t^{\epsilon_n}_{e_n} g_n.  Suppose, say, that e_1 doesn’t originate from v_0.  Then insert stable letters corresponding to edges in \Lambda that form a path from v_0 to the origin of e_1.  Now repeat.