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Normal form theorem for graphs of groups. Let \mathcal{G} be a graph of groups and G=\pi_1(G).

  1. Any g \in G can be written as
    g = g_0 t_{e_1}^{\epsilon_1} g_1 \cdots t_{e_n}^{\epsilon_n} g_n\quad
    as before.
  2. If g = 1, this expression includes `backtracking’, meaning that for some i, e_i = e_{i + 1} with \epsilon_i = - \epsilon_{i + 1}, and furthermore that if \epsilon_i = \pm 1, then g_i \in \partial_\pm ( G_{e_i} ).

Similar to the case of the free group, the proof boils down to the fact that the Bass–Serre tree T is a tree.

Proof. To simplify notation, set t_i = t_{e_i}^{\epsilon_i}, so

g = g_0 t_1 g_1 \cdots t_n g_n.

Fix base points *_i in the vertex spaces X_{v_i}, which are chosen to coincide when the vertices do. Then g_i is a loop in X_{v_i} based at *_i, and t_i is a path, crossing the corresponding edge space, from *_i to *_{i + 1}. This allows us to consider g as a loop in X_{\mathcal{G}} based at *_0. (We may assume v_0 = v_n by adding letters from a maximal tree.)

Consider the universal covering \widetilde{X}_{\mathcal{G}} and fix a base point \widetilde{*}_0 over *_0 in \widetilde{X}_{\widetilde{v}_0}. Let \widetilde{g} be the lift of g based at \widetilde{*}_0 and \gamma its image in the Bass–Serre tree T. We now analyze \widetilde{g} and \gamma closely.

Choose \widetilde{e}_i adjoining \widetilde{X}_{\widetilde{v}_0} and \widetilde{X}_{\widetilde{v}_1} so that the edge traversed by t_i when lifted at \widetilde{*}_{i - 1} corresponds to the coset 1 \cdot G_{e_i} \subseteq G_{v_i} / G_{e_i}.

Then g_0 lifts to a path in \widetilde{X}_{\widetilde{v}_0} which terminates at g_0 \widetilde{*}_0. Similarly, t_1 lifts at *_0 to a path across the edge \widetilde{e}_1 to the vertex space t_1\widetilde{X}_{\widetilde{v}_1} terminating at t_1 \widetilde{*}_1. Therefore, g_0 t_1 lifts at \widetilde{*}_0 to a path which crosses the edge space g_0 \widetilde{e}_1 and ends at g_0 t_1 \widetilde{*}_1.

Then, g_1 lifts at \widetilde{*}_1 to a path in \widetilde{X}_{\widetilde{v}_1} ending at \widetilde{*}_1, and t_2 lifts at \widetilde{*}_1 to a path across the edge \widetilde{e}_2 into the vertex space \widetilde{X}_{\widetilde{v}_2}, and terminating at t_2 \widetilde{*}_2. Thus g_0 t_1 g_1 t_2 lifts at \widetilde{*}_0 to a path which crosses g_0 \widetilde{e}_1, through g_0 \widetilde{X}_{\widetilde{v}_1}, across g_0 t_1 g_1 \widetilde{e}_2, and ending at

g_o t_1 g_1 t_2 \widetilde{*}_2 \in g_0 t_1 g_1 t_2 \widetilde{X}_{\widetilde{v}_2}.

sdoc0001-1

We continue this process until we have explicitly constructed \widetilde{g}. By hypothesis, g = 1, so \widetilde{g} and \gamma are both loops in \widetilde{X}_{\mathcal{G}} and T, respectively. Since T is a tree, \gamma must backtrack.

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This implies that \widetilde{e}_i=\widetilde{e}_{i + 1} and that \epsilon_i = -\epsilon_{i + 1}. That is, by Lemma 18,

g_{i + 1} \in \partial_\pm^{e_i} ( G_{e_i} ).

Therefore, we have found a backtracking, and can accordingly shorten g. This proves the theorem. \square

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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.

Last time we saw that if G acts on a tree, T then G/T has the structure of a graph of groups (remeber stabilizers). Such a \mathcal{G}=G/\Gamma is called developable.

Exercise 18: Show that SL_2\mathbb{Z}\cong(\mathbb{Z}/4\mathbb{Z})\ast_{\mathbb{Z}/2\mathbb{Z}}(\mathbb{Z}/6\mathbb{Z}).

Theorem 13 (Scott-Wall): Let \mathscr{X} be a graph of spaces. The universal cover of X_{\mathscr{X}} is itself a graph of spaces. Also, each vertex space (resp. edge space) is the universal cover of a vertex space of \mathscr{X} (.edge space).

Sketch of Proof: For any v\in V(\Xi), let L_v=X_v\cup(\cup_{e\in E}X_e\times [\pm 1,0]), where E is the set of edges that are the edges incident to v. It should be noted that X_v is a deformation retract of L_v. Also, recall that the edge maps are \pi_1 injective. From covering space theory we know that given a map f:Y\to X and a covering space \hat X of X that f lifts to a map \hat f:Y\to \hat X if and only if f_\ast (\pi_1(Y))\subset \pi_1(\hat X). It therefore follows that \tilde X_v \hookrightarrow \tilde L_v. So \tilde L_v is built from \tilde X_v by attaching covering spaces of edge spaces. Because \partial_{\pm} is \pi_1 injective, these covering spaces of edge spaces really are universal covers. By iteratively gluing together copies of the \tilde L_v we can construct a simply connected cover of X_{\mathscr{X}}.

Given X_{\mathscr{X}} we have constructed the universal cover \tilde X_{\mathscr{X}}. The underlying graph, T of \tilde X_{\mathscr{X}} is a tree because \pi_1(\tilde X_{\mathscr{X}})\to\pi_1(T) is a surjection. We now need to check that the action of G=\pi_1(X_{\mathscr{X}}) on T is interesting.

Let \mathcal{G} be a graph of groups and let G=\pi_1(\mathcal{G}). Fix a base point \ast\in X_v \subset X_{\mathcal{G}}  and a choice of lift \tilde\ast\in \tilde X_{\tilde v}\subset \tilde X_{\mathcal{G}}. Let g\in G_v=\pi_1(X_v).  The space \tilde X_{\tilde v} is a universal cover of X_v by Theorem 13, and so the lift of g to the universal cover of X_{\mathcal{G}} at \tilde\ast is contained in \tilde X_{\tilde v}. Therefore the preimages of \ast that are contained in \tilde X_{\tilde v} correspond to the elements of G_v.

Now consider g\in G\smallsetminus G_v.  If g is lifted at \tilde \ast then the terminus of this lift is not in \tilde X_{\tilde v}, but in some other component of the preimage of X_v.  Call the component where the lift terminates \tilde X_{\tilde v_1}. If g,h\in G are such that both have lifts that terminate in \tilde X_{\tilde v_1} then h^{-1}g\in G_v. We have just proved the following lemma.

Lemma 16: Let \mathcal{G} be a graph of groups and let T be the underlying graph of the universal cover \tilde X_{\mathcal{G}}. For any vertex v\in V(\Gamma) the set of vertices of T lying above v is in bijection with G/G_v and G acts by left translation.

We can also prove the following two lemmas in a similar fashion.

Lemma 17: For any e\in E(\Gamma) the set of edges of T that lie above e is in bijection with G/G_e and G acts by left translation.

Lemma 18: If e\in E(\Gamma) adjoins a vertex v\in V(\Gamma) then for any \tilde v\in V(T) lying above v the set of edges of T adjoining \tilde v lying above e is in bijection wiht G_v/G_e and G_v acts by left translation.