commutator length

Noun

 * 1) The number of multiplicands needed, at a minimum, to express a given group element as a product of commutators.
 * In $$\langle a,b|b^2=1\rangle$$, the commutator length of $$aba^{-2}ba$$ is two, since it equals $$(aba^{-1}b^{-1})(b^{-1}a^{-1}ba)$$ but isn't a commutator.
 * 1) The supremum, over all elements of a given group's derived subgroup, of their commutator lengths.
 * The commutator length of $$\langle a,b|b^2=1\rangle$$ is at least two, since there's an element of commutator length two in it.