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Commutators in Free Groups

We study the representations of an element of a free group as a commutator. For a given element g of a free group F, we are interested in the set of all pairs (x, y) of elements of F such that (1) where [x, y] = xyx-1 y-1. If g = 1, the problem is trivial. We assume henceforth that g ≠ 1.