= Center of a proper amalgam lies in the amalgamated subgroup
If C is proper in both factors of $G=G_1*_CG_2$, 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
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