11.20 (1990)
SolvedSuppose we have $[a, b] = [c, d]$ in an absolutely free group, where $a, b, [a, b]$ are basic commutators (in some fixed free generators). If $c$ and $d$ are arbitrary (proper) commutators, does it follow that $a = c$ and $b = d$?
Progress
No, not always. For example, if $x_1, x_2, x_3$ are free generators, $x_1 < x_2 < x_3$, $a = [[x_2, x_1], x_3]$, $b = [x_2, x_1]$, $c = b^{-1}$, $d = b^{x_3 b}$ (V. G. Bardakov, Abstracts of the IIIrd Intern. Conf. on Algebra, Krasnoyarsk, 1993, p. 33 (Russian)).
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