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 #92;langlea,b#124;b²#61;1#92;rangle, the commutator length of aba#123;-2#125;ba is two, since it equals (aba#123;-1#125;b#123;-1#125;)(b#123;-1#125;a#123;-1#125;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 #92;langlea,b#124;b²#61;1#92;rangle is at least two, since there's an element of commutator length two in it.