Ideals of equations for elements in a free group and context-free languages

Abstract

Let F be a finitely generated free group, and H≤F a finitely generated subgroup. An equation for an element g∈F with coefficients in H is an element w(x)∈H⁎〈x〉 such that w(g)=1 in F; the degree of the equation is the number of occurrences of x and x−1 in the cyclic reduction of w(x). Given an element g∈F, we consider the ideal Ig⊆H⁎〈x〉 of equations for g with coefficients in H; we study the structure of Ig using context-free languages.We describe a new algorithm that determines whether Ig is trivial or not; the algorithm runs in polynomial time. We also describe a polynomial-time algorithm that, given d∈N, decides whether or not the subset Ig,d⊆Ig of all degree-d equations is empty. We provide a polynomial-time algorithm that computes the minimum degree dmin of a non-trivial equation in Ig. We provide a sharp upper bound on dmin.Finally, we study the growth of the number of (cyclically reduced) equations in Ig and in Ig,d as a function of their length. We prove that this growth is either polynomial or exponential, and we provide a polynomial-time algorithm that computes the type of growth (including the degree of the growth if it's polynomial).