A general method for calculating or constructing lower central factors of groups is presented. Relative basic commutators are defined.

It is well known that the set of values a lower central word in group G need not be subgroup. For fixed ?r and for p ? 5, Guralnick showed if finite p-group such verbal subgroup ?r(G) abelian 2-generator, then consists only ?r-values. In this paper we extend result, showing assumption can dropped. Moreover, show result remains true even p= 3. Finally, prove analogous pro-p groups true.

We show that the Vassiliev invariants of a knot K, are obstructions to finding a regular Seifert surface, S, whose complement looks “simple” (e.g. like the complement of a disc) to the lower central series of its fundamental group.

We study a class of positively graded Lie algebras with a pattern of homogeneous components similar to that of the graded Lie algebra associated to the Nottingham group with respect to its lower central series.

We determine the function fields of varieties of dimension ≥ 2 defined over the algebraic closure of Fp, modulo purely inseparable extensions, from the quotient by the second term in the lower central series of their pro-` Galois groups.

The terms of the upper and lower central series of a nilpotent computable group have computably enumerable Turing degree. We show that the Turing degrees of these terms are independent even when restricted to groups which admit computable orders.