2022/07/11 by Ascari, Dario · 1 citation
#20E05 (20F65) #20F70 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2207.04759
Let F be a finitely generated free group and let H≤ F be a finitely generated subgroup. Given an element g∈ F, we study the ideal \mathfrakIg of equations for g with coefficients in H, i.e. the elements w(x)∈ H*⟨ x⟩ such that w(g)=1 in F. The ideal \mathfrakIg is a normal subgroup of H*⟨ x⟩, and we provide an algorithm, based on Stallings folding operations, to compute a finite set of generators for \mathfrakIg as a normal subgroup. We provide an algorithm to find an equation in \mathfrakIg 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∈ℕ, determines whether \mathfrakIg 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 Dg of all integers d such that \mathfrakIg contains equations of degree d; we show that Dg 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 rank(H)=1. We prove that both kinds of sets Dg can actually occur. The examples also show that the equations of minimum possible degree aren't in general enough to generate the whole ideal \mathfrakIg as a normal subgroup.