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Paper   IPM / M / 7704
School of Mathematics
  Title:   Products of communicators in free groups
1.  M. Akhavan-Malayeri
2.  A. Rhemtulla
  Status:   Published
  Journal: Internat. J. Algebra Comput.
  Vol.:  13
  Year:  2004
  Pages:   231-240
  Supported by:  IPM
Let F be the free group of rank 2, generated by x and y, and let w(x,y) ∈ F′ be a nontrivial 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 ∈ (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 ranks are also presented.

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