Normal form theorem for an amalgamated free product

In $A{\ast }_{C}B$, an alternating product of elements of $A\setminus C$ and $B\setminus C$ is nonidentity. More generally, choosing coset representatives gives a unique reduced normal form.

This is the amalgamated case of normal form for free groups and free product of groups.

If C is proper in both factors of $G=G_1{\ast }_C{G}_2$, every central element lies in C. In the Bass-Serre tree, a central element fixing a vertex fixes its entire orbit and then the whole tree, hence an edge. A central element acting as a translation would have an invariant axis; minimality forces the tree to be that axis, but a vertex stabilizer then contains a reflection, contradicting centrality of the translation. Thus the center fixes an edge and is contained in its stabilizer C.

// Target: algebraic-topology.bigb