  The quotients $G_k/G_{k+1}$ of the lower central series of a finitely
presented group $G$ are an important invariant of this group. In this work we
investigate the ranks of these quotients in the case of a certain class of
conjugation-free groups, which are groups generated by $x_1,...,x_n$, and
having only cyclic relations: $$ x_{i_t} x_{i_{t-1}} ... x_{i_1} = x_{i_{t-1}}
... x_{i_1} x_{i_t} = ... = x_{i_1} x_{i_t} ... x_{i_2}.$$ Using tools from
group theory and from the theory of line arrangements we explicitly find these
ranks, which depend only at the number and length of these cyclic relations. It
follows that for these groups the associated graded Lie algebra $gr(G)$
decomposes, in any degree, as a direct product of local components.
