PRODUCTS OF COMMUTATORS IN FREE GROUPS
Mehri Akhavan-Malayeri, A. H. Rhemtulla · International Journal of Algebra and Computation · 2003
Let F be the free group of rank 2, generated by x and y, and let w(x, y) ∈ F′ be a non-trivial word. We give elementary algebraic proofs and algorithms to (1) express [x, y]n as a product of [n/2] + 1 commutators and show this is the best possible; (2) show that (w(x, y))2 cannot be written as one w-word and if g ≠ 1 ∈ w(F) then show that the minimal number of w-words required to express gn as their product tends to infinity with n. Other results for free groups of higher rank are also presented.