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IPM
30
YEARS OLD

“School of Mathematics”

Paper   IPM / M / 7704
   School of Mathematics
  Title: Products of communicators in free groups
  Author(s):
1 . M. Akhavan-Malayeri
2 . A. Rhemtulla
  Status: Published
  Journal: Internat. J. Algebra Comput.
  Vol.: 13
  Year: 2004
  Pages: 231-240
  Supported by: IPM
  Abstract:
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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