Effective Aspects of Grushko's Theorem: A Tribute to John Stallings, 1935 - 2008

In his 1959 Princeton Ph. D. dissertation, John Stallings gave a topological proof of a 1940 theorem of I. Grushko about generating sets of free products of groups. Stallings' proof is actually a method that has "a certain effectiveness" based on a subtle combinatorial fact that I will describe. As inspiration for his proof, after the fact, Stallings cited a 1929 paper of H. Kneser on decompositions of three-manifolds. Stallings went on to pioneer theories about decompositions of infinite groups, which reamins an active area of research.