Finitely generated simple groups of infinite commutator width.

3 March 2008
Alex Muranov
If $G$ is a group and $g$ an element of the derived subgroup $[G,G]$, the commutator length of $g$ is the least positive integer $n$ such that $g$ can be written as a product of $n$ commutators. The commutator width of $G$ is the maximum of the commutator lengths of elements of $[G,G]$. Until 1991, to my knowledge, it has not been known whether there exist simple groups of commutator width greater than $1$. The same question for finite simple groups still remains unsolved. In 1992, Jean Barge and √Čtienne Ghys showed that the commutator width of certain simple groups of diffeomorphisms is infinite. However, those groups are not finitely generated. Finitely generated infinite simple groups of infinite commutator width can be constructed using "small cancellations." Additionally, finitely generated infinite boundedly simple groups of arbitrarily large (but necessarily finite) commutator width can be constructed in a similar way.