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I've wondered for a while about the (Hall-)Witt identity in group theory:

$[[a,b^{-1}],c]^b \cdot [[b,c^{-1}],a]^c \cdot [[c,a^{-1}]],b]^a = 1$.

(Here, $x^y$ means $y^{-1}xy$ and $[x,y]$ means $x^{-1}y^{-1}xy$.) Does anybody have any motivation for this? To me, it almost seems like it comes out of nowhere so that we can prove the three subgroup lemma or something. Is there some reason to expect a relation like this to hold, or a way of reducing it to simpler relations in a meaningful way? Perhaps we should expect something like this from the free-ness of the commutator subgroup of the free group on three letters? Or should we expect some analogue of the Jacobi identity to hold, and if so, why?

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    $\begingroup$ It may be intersting to note that if $G$ is metaablian (i.e. the derived subgroup $[G,G]$ is abelian), then the Hall-Witt identity looks like exactly as the Jacobi identity, that is to say you may ignore and drop the conjugates and the inverses in the identity; That is $$[a,b,c][b,c,a][c,a,b]=1,$$ holds for all $a,b,c\in G$ whenever $G$ is metaabelian. $\endgroup$
    – Alireza Abdollahi
    Commented Nov 18, 2011 at 7:23
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    $\begingroup$ Well... the motivation for introducing it might be that 1) it looks like the Jacobi identity 2) from it, you can derive the Jacobi identity for the Lie algebra of an algebraic group 3) from it, you can derive the Jacobi identity for the Lie algebra associated to the lower central series of a discrete group. $\endgroup$
    – André Henriques
    Commented Nov 18, 2011 at 8:46
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    $\begingroup$ When it comes to such identities in general, what you are really trying to understand is the structure of free groups. The theory of free groups has a strong geometric flavour, so perhaps there is a geometric interpretation of the Hall-Witt identity that makes it seem more natural. $\endgroup$
    – Colin Reid
    Commented Nov 18, 2011 at 16:54
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    $\begingroup$ lamington.wordpress.com/2011/11/20/the-hall-witt-identity $\endgroup$
    – Terry Tao
    Commented Nov 20, 2011 at 17:37
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    $\begingroup$ A two variable version of Hall-Witt is the nearly trivial identity [a,b][b,a] = 1. Hall-Witt, of course, has three variables. This suggests the question of the existence of some four variable version. I wonder if some kind of computer search might yield an identity of this kind. $\endgroup$
    – Marty Isaacs
    Commented Jan 25, 2012 at 22:04

1 Answer 1

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To add to Terry Tao's excellent description of some of the geometric ideas behind the Hall-Witt identity, I would draw attention to the interpretation given by Loday in his paper on Homotopical Syzygies (in Une dégustation topologique: Homotopy theory in the Swiss Alps, volume 265 of Contemporary Mathematics, 99 – 127, AMS, 2000; book on AMS site). In his section 1.4. in examining the syzygies / identities amongst the relations of the obvious presentation of the free Abelian group on three symbols, he shows how the commutators/ 2-syzygies are drawn as the faces of the Cayley graph considered as being embedded in a sphere (and thus appearing as an empty cube). There is a 3-syzygy needed to build the next stage of a resolution of the group and that is a 3-cell filling the cube. The boundary of that 3-cell can be read off as the Jacobi-Witt-Hall (JWH) identity (with slightly different conventions more as in Terry's article than in the original form in the question).

Thus the proof of freeness of the commutator subgroup of $F(x,y,z)$ does not give a useful basis for that subgroup and the commutators and their conjugates are not without relations between them. The JWH identity is one of those relations. The exploration of homotopical syzygies throws up some lovely calculations in general (and lots of nice pictures!) (Edit: I mean for quite general presentations.)

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    $\begingroup$ @Y. Cor: Thanks for adding that link, $\endgroup$
    – Tim Porter
    Commented Mar 17, 2018 at 12:07
  • $\begingroup$ I've developed the answer of Tim Porter a bit in the preprint Some commutator identities $\endgroup$
    – Graham Ellis
    Commented Jan 4 at 0:40
  • $\begingroup$ @GrahamEllis That looks very nice. $\endgroup$
    – Tim Porter
    Commented Jan 5 at 6:52
  • $\begingroup$ For the record, the "excellent description" that Terry Tao linked to was written by Danny Calegari. $\endgroup$
    – HJRW
    Commented Jan 5 at 14:02

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