Last time we saw that if acts on a tree, then has the structure of a graph of groups (remeber stabilizers). Such a is called developable.
Exercise 18: Show that .
Theorem 13 (Scott-Wall): Let be a graph of spaces. The universal cover of is itself a graph of spaces. Also, each vertex space (resp. edge space) is the universal cover of a vertex space of (.edge space).
Sketch of Proof: For any , let , where is the set of edges that are the edges incident to . It should be noted that is a deformation retract of . Also, recall that the edge maps are injective. From covering space theory we know that given a map and a covering space of that lifts to a map if and only if . It therefore follows that . So is built from by attaching covering spaces of edge spaces. Because is injective, these covering spaces of edge spaces really are universal covers. By iteratively gluing together copies of the we can construct a simply connected cover of .
Given we have constructed the universal cover . The underlying graph, of is a tree because is a surjection. We now need to check that the action of on is interesting.
Let be a graph of groups and let . Fix a base point and a choice of lift . Let . The space is a universal cover of by Theorem 13, and so the lift of to the universal cover of at is contained in . Therefore the preimages of that are contained in correspond to the elements of .
Now consider . If is lifted at then the terminus of this lift is not in , but in some other component of the preimage of . Call the component where the lift terminates . If are such that both have lifts that terminate in then . We have just proved the following lemma.
Lemma 16: Let be a graph of groups and let be the underlying graph of the universal cover . For any vertex the set of vertices of lying above is in bijection with and acts by left translation.
We can also prove the following two lemmas in a similar fashion.
Lemma 17: For any the set of edges of that lie above is in bijection with and acts by left translation.
Lemma 18: If adjoins a vertex then for any lying above the set of edges of adjoining lying above is in bijection wiht and acts by left translation.