Normal form for the fundamental group of a graph of groups

If $X$ has no geometric edges, then ${\pi }_1(\mathcal{G},T)$ is the unique vertex group, so every element already has edge length $0$ and the identity clause is immediate. If $X$ has one geometric edge, then either that edge lies in $T$, in which case [L1] and [L2] present ${\pi }_1(\mathcal{G},T)$ as the corresponding amalgamated free product and [L4] gives the required normal form, or it lies outside $T$, in which case [L1] and [L2] present ${\pi }_1(\mathcal{G},T)$ as the corresponding HNN extension and [L5] gives the required normal form.

Assume the theorem holds for graphs of groups with at most $n$ geometric edges, and let $X$ have $n+1$. If $X$ has an edge $e$ outside $T$, remove that edge. The graph $X\setminus \{e\}$ stays connected because it still contains the spanning tree $T$, and [L1] and [L2] identify ${\pi }_1(\mathcal{G},T)$ with the HNN extension of ${\pi }_1(\mathcal{G}{\mathrm{\mid }}_{X\setminus \{e\}},T)$ having stable letter $e$ and associated subgroups the two images of $G_e$. Otherwise $X=T$ is a tree. Choose a leaf edge $e$ of $T$; removing it splits $X$ into connected components $X_1$ and $X_2$, and $T\setminus \{e\}$ restricts to maximal subtrees $T_1\subseteq X_1$ and $T_2\subseteq X_2$. Unwinding [L1] and [L2] then identifies ${\pi }_1(\mathcal{G},T)$ with the amalgamated free product of ${\pi }_1(\mathcal{G}{\mathrm{\mid }}_{X_1},T_1)$ and ${\pi }_1(\mathcal{G}{\mathrm{\mid }}_{X_2},T_2)$ over $G_e$.

In each smaller graph-of-groups fundamental group, the induction hypothesis supplies reduced closed representatives. Applying the amalgam normal form [L4] or the HNN normal form [L5] to the decomposition from step 1.2 and then expanding the smaller representatives back into the original generators gives a closed graph-of-groups word in which the only forbidden simplifications are exactly the backtracking patterns ruled out by [L3]. Hence every element of ${\pi }_1(\mathcal{G},T)$ has a reduced closed graph-of-groups representative.

The same normal-form theorems [L4] and [L5] say that a reduced amalgam word or Britton-reduced word is nonidentity whenever it has positive syllable length, and that syllable length $0$ gives the identity only when the remaining base-group coefficient is the identity. Under the identifications of step 1.2, this is exactly the statement that a reduced closed graph-of-groups word of positive edge length is nonidentity, and that an edge-length-$0$ reduced closed word is the identity only when its unique coefficient is the identity in the relevant vertex group.