Abstract. Let F be a finite-rank free group and H be a finite-rank subgroup of F . We discuss proofs of two algorithms that sandwich H between an upper-layer free-product factor of F that contains H and a lower-layer free-product factor of F that is contained in H.Richard Stong showed that the unique smallest-possible upper layer, denoted Cl(H), is visible in the output of the polynomial-time cut-vertex algorithm of J. H. C. Whitehead. Stong's proof used bi-infinite paths in a Cayley tree and sub-surfaces of a three-manifold. We give a variant of his proof that uses edge-cuts of the Cayley tree induced by edge-cuts of a Bass-Serre tree.A. Clifford and R. Z. Goldstein gave an exponential-time algorithm that determines whether or not the trivial subgroup is the only possible lower layer. Their proof used Whitehead's three-manifold techniques. We give a variant of their proof that uses Whitehead's cut-vertex results, and thereby obtain a somewhat simpler algorithm that yields a lower layer of maximum-possible rank.2010 Mathematics Subject Classification. Primary: 20E05; Secondary: 20E36, 20E08. Key words. Sub-bases of free groups. Free-product factors. Cut-vertex algorithm. Cut-vertex lemma. Clifford-Goldstein algorithm.1 Introduction 1.1 Definitions. For any set E, we let E | denote the free group on E. By a basis of E | , we mean a free-generating set of E | . By a sub-basis of E | we mean a subset of a basis of E | . We let Aut E | denote the group of automorphisms of E | acting on the right as exponents.For any subset Z of E | , we let Z denote the subgroup of E | generated by Z. We let supp(Z rel E) denote the ⊆-smallest subset of E such that Z ⊆ supp(Z rel E) . We let Cl(Z) denote the intersection of all the free-product factors (generated by sub-bases) of E | that contain Z. * Partially supported by Spain's Ministerio de Ciencia e Innovación through Project MTM2011-25955.
2Sandwiching between free-product factors 1.2 Hypotheses. Throughout, let E be a finite set, let Z be a finite subset of E | , and let E Z denote supp(Z rel E).1.3 History. Recall Hypotheses 1.2.• In [8, publ. 1936], J. H. C. Whitehead gave his true-word and cyclic-word cut-vertex algorithms, and the former determines whether or not Z is a sub-basis of E | . A little later, in [9, publ. 1936], he gave an exponential-time, general-purpose algorithm which has largely overshadowed the easier-to-prove, polynomial-time, limited-use algorithm. We wish to emphasize that the cutvertex algorithm suffices to efficiently sandwich a subgroup between two freeproduct factors.Whitehead defined a certain finite graph which we denote Wh * (Z rel E Z ). He observed that if some vertex of Wh * (Z rel E Z ) is what we call a Whitehead cut-vertex, then it is straightforward to construct an automorphism of E | that strictly reduces the total E-length of Z. Clearly, one then has an algorithm (with choices) which constructs some Ψ ∈ Aut E | such that Wh * (Z Ψ rel E Z Ψ ) has no Whitehead cut-vertices. It then remains to extract information from Ψ and Z Ψ . For ex...
