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Paper IPM / M / 7704  


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 wword and if g ≠ 1 ∈ (F) then show that the
minimal number of wwords required to express g^{n} as their
product tends to infinity with n. Other results for free groups
of higher ranks are also presented.
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