Topology of Finite Graphs

This paper derives from a course in group theory which I gave at Berkeley in 1982. I wanted to prove the standard theorems on free groups, and discovered that, after a few preliminaries, the notion of "locally injective" map (or "immersion") of graphs was very useful. This enables one to see, in an effective, easy, algorithmic way just what happens with finitely generated free groups. One can understand in this way (1) Howson's theorem that if A and B are finitely generated subgroups of a free group, then A ~ B is finitely generated, and (2) M. Hall's theorem that finitely generated subgroups of free groups are closed in the profinite topology. During this course, S.M. Gersten came up with a simple proof of H. Neumann's inequality on the ranks in Howson's theorem. One of the ideas in Gersten's proof was to use core-graphs (graphs with no trees hanging on). Subsequently, I found that some consequences of a paper of Greenberg's could be proved using core-graphs and "covering translations" of immersions; the most striking such result is that if A and B are finitely generated subgroups of a free group and if A c~ B is of finite index in both A and B, then A ~ B is of finite index in A v B, the subgroup generated by A uB.