2019/07/17 by Duncan, Andrew J., Juhász, Arye
#20E06 #20F05 #20F36 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.1907.07797
We generalise a key result of one-relator group theory, namely Magnus's Freiheitssatz, to partially commutative groups, under sufficiently strong conditions on the relator. The main theorem shows that under our conditions, on an element r of a partially commutative group \mathbbG, certain Magnus subgroups embed in the quotient G=\mathbbG/N(r); that if r=sn has root s in \mathbbG 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 \mathbbG embed in G reduces to the same question in the minimal parabolic subgroup of \mathbbG 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 \mathbbG, 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 partially commutative group \mathbbG(Γ) satisfy the conditions of the theorem.