commutator lengths

Noun

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.