A proof of Serre's theorem (Q1812759)

In 1869 Jordan observed that a group of automorphisms of a finite tree must fix a vertex or an edge. During the course of the next hundred years it became clear that, while the result does not extend to infinite trees, the structure of a group of automorphisms acting without fixed vertices or edges on an infinite tree is very special. The theorem that such a group is free was formulated by \textit{J.-P. Serre}. The proof given in his book [Trees. Transl. from the French (1980; Zbl 0548.20018){]} is clearly motivated by topological considerations; it involves the construction of a quotient graph, lifting a spanning tree, contracting the images of the spanning tree, and proving that the shrunken graph is a tree isomorphic to a Cayley graph. \textit{D. E. Cohen} [Combinatorial group theory. A topological approach (1978; Zbl 0389.20024){]} asserts that the outline of this proof was given by {Reidemeister} [Einführung in die kombinatorische Topologie (1932; Zbl 0004.36904)], and he shows that some of the topological machinery is unnecessary; but his proof requires the same basic steps. The object of the present paper is to give a straightforward proof in the locally finite case, using only the facts that a tree is an acyclic connected graph and that it has a natural metric preserved by automorphisms. The proof is not fundamentally new, but it does avoid several extraneous constructions. Also, it is formulated within the ordinary terminology of graph theory.