Cyclically presented groups, lower central series and line arrangements

The quotients Gk/Gk+1Gk/Gk+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 cyclically presented groups, which are groups generated by x1,…,xnx1,…,xn and having only cyclic relations:xitxit−1⋅…⋅xi1=xit−1⋅…⋅xi1xit=⋯=xi1xit⋅…⋅xi2.xitxit−1⋅…⋅xi1=xit−1⋅…⋅xi1xit=⋯=xi1xit⋅…⋅xi2.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)gr(G) decomposes, in any degree, as a direct product of local components.