Commutators and the Ore Conjecture

In a talk yesterday by Boris Kunyavski at the University of Ottawa, I learned a little about the Ore conjecture, which in 2010 was proved a theorem in:

Liebeck, Martin W.; O’Brien, E. A.; Shalev, Aner; Tiep, Pham Huu. The Ore conjecture. J. Eur. Math. Soc. (JEMS) 12 (2010), no. 4, 939-1008.

It’s quite a fascinating result that arises by considering commutators in groups. If $G$ is a group, its commutator subgroup $[G,G]$ is the subgroup of $G$ generated by all the commutators $[g,h] = ghg^{-1}h^{-1}$ of $G$. It’s easy to see that the commutator subgroup is normal. A group $G$ is said to be perfect if $G = [G,G]$.

So let’s assume $G$ is perfect. This implies that every element of $G$ can be written as a product of commutators. But can every element of $G$ be written as a single commutator? That’s really far from obvious. For example, take your favourite perfect group and an element in it: can you prove that this single element is a commutator? Not so easy, right?

In fact, we can define the commutator length of any $g\in G$ to be the minimum number of commutators in all products of commutators equal to $g$. If $g$ can’t be written as the product of commutators, then its commutator length is infinite.

The commutator width of a group is defined to be the supremum over commutator lengths of all the elements of $G$. (Note: I think this should just be called the commutator length of $G$ as well, but that’s how the terminology ended up!)

It turns out that finding a perfect group $G$ with commutator width greater than one is quite tricky. In fact, the theorem proved in loc. cit. is:

Former Ore Conjecture/Now Theorem. If $G$ is a finite nonabelian simple group, then every element of $G$ is a commutator.

That’s pretty cool, though the proof is very long. That’s not surprising since it is a theorem about all finite nonabelian simple groups. What’s perhaps even more surprising is that there are examples of finitely-generated infinite simple groups containing elements that are not commutators. In fact, examples exist of such $G$ with infinite commutator length, as given in Alexey Muranov’s paper:

Finitely generated infinite simple groups of infinite square width and vanishing stable commutator length. J. Topol. Anal. 2 (2010), no. 3, 341-384.

This result makes use of small cancellation theory, which is a geometric group theory machinery that studies presented groups whose relations don’t have too much in common.

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