One-relator quotients of right-angled Artin groups

Abstract

We generalise a key result of one-relator group theory, namely Magnus's Freiheitssatz, to right-angled Artin groups, under sufficiently strong conditions on the relator. The main theorem shows that under our conditions, on an element r of a right-angled Artin group G, certain Magnus subgroups embed in the quotient G=G/N(r); that if r=sn has root s in G then the order of s in G is n, and under slightly stronger conditions that the word problem of G is decidable. We also give conditions under which the question of which Magnus subgroups of G embed in G reduces to the same question in the minimal parabolic subgroup of G containing r. In many cases this allows us to characterise Magnus subgroups which embed in G, via a condition on r and the commutation graph of G, and to find further examples of quotients G where the word and conjugacy problems are decidable. We give evidence that situations in which our main theorem applies are not uncommon, by proving that for cycle graphs with a chord Γ, almost all cyclically reduced elements of the right-angled Artin group G(Γ) satisfy the conditions of the theorem.