Dependence over subgroups of free groups

Abstract

Given a finitely generated subgroup H of a free group F, we present an algorithm which computes [Formula: see text], such that the set of elements [Formula: see text], for which there exists a non-trivial H-equation having g as a solution is precisely the disjoint union of the double cosets [Formula: see text]. Moreover, we present an algorithm which, given a finitely generated subgroup [Formula: see text] and an element [Formula: see text], computes a finite set of elements from [Formula: see text] (of the minimum possible cardinality) generating, as a normal subgroup, the “ideal” [Formula: see text] of all “polynomials” [Formula: see text], such that [Formula: see text]. The algorithms, as well as the proofs, are based on the graph-theoretic techniques introduced by Stallings and on the more classical combinatorial techniques of Nielsen transformations. The key notion here is that of dependence of an element [Formula: see text] on a subgroup H. We also study the corresponding notions of dependence sequence and dependence closure of a subgroup.