Basic Commutators in Weights Six and Seven as Relators

Charles Sims has asked whether or not the lower central subgroup γ _n (F) of a free group F coincides with the normal closure in F of the basic commutators of weight n. This question has a positive answer in weights at most 5, but remains an open question in general. In earlier work with Gaglione and Spellman, it was shown that γ _n (F) is the normal closure in F of the basic commutators of weights n through 2n ? 4. Here, we specialize to the case where F has rank 2 and show that γ₆(F) is the normal closure in F of the basic commutators of weights 6 and 7.