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We write $cl$ for the commutator length, i.e. the least number of commutators which multiply to a given element of a group.

Given an element $g$ in the commutator subgroup of the free group $G=F_2$ on two generators, is it true that $$cl_G(g) = \displaystyle \max_{\mbox{H < G finite index normal}} cl_H cl_{G/H} (g \mod H)$$ ?

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# Commutator length modulo finite index subgroups

We write $cl$ for the commutator length, i.e. the least number of commutators which multiply to a given element of a group.

Given an element $g$ in the commutator subgroup of the free group $G=F_2$ on two generators, is it true that $$cl_G(g) = \displaystyle \max_{\mbox{H < G finite index normal}} cl_H (g \mod H)$$ ?