On free-group algorithms that sandwich a subgroup between free-product factors

Let F be a finite-rank free group and Z be a finite subset of F. We give topology-free proofs for two algorithms that yield sub-bases E″ and E′ of F satisfying 〈E″〉 ≤ 〈Z〉i ≤ 〈E′〉 that minimize the value |E′| - |E′|. Here, the subgroup 〈E′〉 is uniquely determined, and Richard Stong showed that a special basis thereof is produced by J. H. C. Whitehead's cut-vertex algorithm. Stong's proof used bi-infinite paths in a Cayley tree and sub-surfaces of a handlebody. We give a new proof that uses edge-cuts of the Cayley tree that are induced by edge-cuts of a Bass Serre tree. A. Clifford and R. Z. Goldstein used Whitehead's three-manifold techniques to give an algorithm that determines whether or not there exists a basis of F that meets 〈Z〉. We replace the topology with the cut-vertex algorithm, and obtain a slightly simpler Clifford Goldstein algorithm that yields a basisB of F that maximizes the value |B∩〈Z〈|. © de Gruyter 2014.