Topological groups, automorphisms of infinite graphs and a theorem of Trofimov (Q1377826)

This paper gives a short proof of a theorem of Trofimov about locally finite infinite graphs. The theorem is that the subgroup of bounded automorphisms is vertex-transitive if and only if the graph has a system of imprimitivity with finite blocks such that the group induced on the blocks is a free finitely generated abelian group. Here a bounded automorphism means one having a bound on the distance (in the graph) between any vertex and its image. The short proof in the present paper uses some lemmas about compact subsets in topological groups.