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.