commutator length

commutator length (plural commutator lengths)

1. The number of multiplicands needed, at a minimum, to express a given group element as a product of commutators.

   In ${\displaystyle \langle a,b|b^{2}=1\rangle }$, the commutator length of ${\displaystyle aba^{-2}ba}$ is two, since it equals ${\displaystyle (aba^{-1}b^{-1})(b^{-1}a^{-1}ba)}$ but isn't a commutator.

2. The supremum, over all elements of a given group's derived subgroup, of their commutator lengths.

   The commutator length of ${\displaystyle \langle a,b|b^{2}=1\rangle }$ is at least two, since there's an element of commutator length two in it.

• stable commutator length