Lemma 7

We have one of the following:

a) X has a unique ramification point (Chap. IV, Appendix, no. 1). and all the edges of X are of order 3. b) X is a chain and has at most one edge of order $\geq 4$.

We argue by induction on l.

a) Assume that X has a ramification point i. Then i belongs to 3 edges of order 3. $\{i, k_1\}, \{i, k_2\}, \{i, k_3\}$ (Lemmas 2 and 3). If $l = 4$ the lemma is proved. If not, then $k_1$, say, belongs to an edge distinct from those just mentioned since X is connected. Identify i and $k_1$ in the Coxeter graph of M. This gives a new graph to which the induction hypothesis can be applied, in view of Lemma 6. Now the image $p$ of i is a ramification point of the new graph $X'$. Thus $X'$ has no other ramification point and all its edges are of order 3. Thus all the edges of X are of order 3, and i and $k_1$ are its only possible ramification points. But if $k_1$ were a ramification point of X, $p$ would belong to at least 4 edges in $X'$, contrary to Lemma 2.

b) Assume that X has no ramification point. Then X is a chain (Chap. IV, Appendix, no. 3, Prop. 3). Let $\{i, j\}$ be an edge of order $\geq 4$. If $l = 2$, the lemma is trivial. If not, then i, say, belongs to an edge $\{i, k\}$ with $k \neq j$ (since X is connected). This edge is of order 3 (Lemma 5). Identify i and k in the Coxeter graph of M. By Lemma 6, the induction hypothesis can be applied. Let $p$ be the image of i in the new graph $X'$. In $X'$, $\{p, j\}$ is an edge of order $\geq 4$, so $X'$ has no other edge of order $\geq 4$, and hence $\{i, j\}$ is the only edge of order $\geq 4$ in X.