Normal form theorem for graphs of groups. Let be a graph of groups and .
- Any can be written as
- If , this expression includes `backtracking’, meaning that for some , with , and furthermore that if , then .
Similar to the case of the free group, the proof boils down to the fact that the Bass–Serre tree is a tree.
Proof. To simplify notation, set , so
Fix base points in the vertex spaces , which are chosen to coincide when the vertices do. Then is a loop in based at , and is a path, crossing the corresponding edge space, from to . This allows us to consider as a loop in based at . (We may assume by adding letters from a maximal tree.)
Consider the universal covering and fix a base point over in . Let be the lift of based at and its image in the Bass–Serre tree . We now analyze and closely.
Choose adjoining and so that the edge traversed by when lifted at corresponds to the coset .
Then lifts to a path in which terminates at . Similarly, lifts at to a path across the edge to the vertex space terminating at . Therefore, lifts at to a path which crosses the edge space and ends at .
Then, lifts at to a path in ending at , and lifts at to a path across the edge into the vertex space , and terminating at . Thus lifts at to a path which crosses , through , across , and ending at
We continue this process until we have explicitly constructed . By hypothesis, , so and are both loops in and , respectively. Since is a tree, must backtrack.
This implies that and that . That is, by Lemma 18,
Therefore, we have found a backtracking, and can accordingly shorten . This proves the theorem.