Basic Commutators in Weights Six and Seven as Relators

Abstract

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.

Key Words:

• Basic commutator
• Free group

2000 Mathematics Subject Classification:

• Primary 20F05, 20F12, 20F18

Notes

Communicated by M. R. Dixon.