Ideals of equations for elements in a free group and Stallings folding

Abstract

Let $F$ be a finitely generated free group and let $H\le F$ be a finitely generated subgroup. Given an element $g\in F$, we study the ideal $\mathfrak{I}_g$ of equations for $g$ with coefficients in $H$, i.e. the elements $w(x)\in H*\langle x\rangle$ such that $w(g)=1$ in $F$. The ideal $\mathfrak{I}_g$ is a normal subgroup of $H*\langle x\rangle$, and we provide an algorithm, based on Stallings folding operations, to compute a finite set of generators for $\mathfrak{I}_g$ as a normal subgroup. We provide an algorithm to find an equation in $\mathfrak{I}_g$ with minimum degree, i.e. an equation $w(x)$ such that its cyclic reduction contains the minimum possible number of occurrences of $x$ and $x^{-1}$; this answers a question of A. Rosenmann and E. Ventura. More generally, we provide an algorithm that, given $d\in\mathbb{N}$, determines whether $\mathfrak{I}_g$ contains equations of degree $d$ or not, and we give a characterization of the set of all the equations of that specific degree. We define the set $D_g$ of all integers $d$ such that $\mathfrak{I}_g$ contains equations of degree $d$; we show that $D_g$ coincides, up to a finite set, either with the set of non-negative even numbers or with the set of natural numbers. Finally, we provide examples to illustrate the techniques introduces in this paper. We discuss the case where $\text{rank}(H)=1$. We prove that both kinds of sets $D_g$ can actually occur. The examples also show that the equations of minimum possible degree aren't in general enough to generate the whole ideal $\mathfrak{I}_g$ as a normal subgroup.