[Paper Review] More on five commutator identities

Гурам Донадзе, M. Ladra|ArXiv.org|Mar 21, 2007

Geometric and Algebraic Topology9 references3 citations

TL;DR

This paper proves Ellis's conjecture that the five universal commutator identities—covering weight-4 commutators in groups—generate all possible universal relations among such commutators. Using multiplicative Lie algebras and nonabelian tensor products, the authors establish that the surjection from the free multiplicative Lie algebra to the lower central series is an isomorphism for weight 4, confirming the conjecture for n=4.

ABSTRACT

It is proved that the five well-known identities universally satisfied by commutators in a group generate all universal commutator identities for commutators of weight 4.

Motivation & Objective

• To verify Ellis's conjecture that the five standard commutator identities generate all universal relations among commutators of weight 4 in any group.
• To extend Ellis's earlier proof for n=2 and n=3 to the case n=4 using homological algebra and nonabelian tensor product techniques.
• To establish that the natural surjection from the free multiplicative Lie algebra to the lower central series is an isomorphism for weight 4.
• To demonstrate that the relations in the nonabelian tensor product $[P,P]_{ ext{ab}} \otimes P_{ ext{ab}}$ capture all universal identities for weight-4 commutators.
• To provide a structural characterization of the kernel of the map $\theta_4$ in terms of generators derived from the five fundamental identities.

Proposed method

• Formalize the structure of multiplicative Lie algebras using five axioms: skew-symmetry, conjugation invariance, and the Hall-Witt-type identity.
• Define $\mathcal{L}(P)$ as the free multiplicative Lie algebra on a group $P$, with $\Gamma_n(P)$ generated by iterated commutators of weight $n$.
• Use the nonabelian tensor product $[P,P]_{\text{ab}} \otimes P_{\text{ab}}$ to model relations among weight-4 commutators, leveraging its abelian structure.
• Construct a homomorphism $\theta_3^{-1} \widetilde{\otimes} P: \gamma_3(P) \otimes P \to \Gamma_4(P)/\Gamma_4'(P)$ to analyze the kernel of $\widetilde{\theta_4}$.
• Prove injectivity of $\widetilde{\theta_4}$ by showing that the kernel is generated by images of specific tensor relations, including $[\gamma_2(P),\gamma_2(P)] \otimes P$ and $\gamma_3(P) \otimes \gamma_2(P)$.
• Use congruences modulo $\Gamma_4'(P)$ and the Hall-Witt identity to reduce complex commutator expressions to trivial elements, confirming that key relations lie in $\Gamma_4'(P)$.

Experimental results

Research questions

• RQ1 Do the five standard universal commutator identities generate all universal relations among commutators of weight 4 in any group?
• RQ2 Is the natural surjection $\theta_4: \Gamma_4(P) \to \gamma_4(P)$ an isomorphism when $P$ is a free group?
• RQ3 Can the kernel of the map $\theta_4$ be fully described using the five fundamental identities and their consequences?
• RQ4 How do nonabelian tensor products $[P,P]_{\text{ab}} \otimes P_{\text{ab}}$ encode the universal relations for weight-4 commutators?
• RQ5 Are the relations $[y, {}^{y^{-1}}[x,y]] \otimes x$ and $[^{y^{-1}}[x,y], x] \otimes {}^x y$ trivial modulo the subgroup $\Gamma_4'(P)$?

Key findings

• The five standard commutator identities generate all universal relations among commutators of weight 4 in any group.
• The surjection $\theta_4: \Gamma_4(P) \to \gamma_4(P)$ is an isomorphism when $P$ is a free group, confirming Ellis’s conjecture for $n=4$.
• The kernel of $\widetilde{\theta_4}: \Gamma_4(P)/\Gamma_4'(P) \to \gamma_4(P)/[\gamma_3(P), \gamma_2(P)]$ is trivial, implying $\widetilde{\theta_4}$ is injective.
• The map $\theta_3^{-1} \widetilde{\otimes} P$ is well-defined and surjective, and its kernel corresponds exactly to the relations needed to kill the cokernel of $\widetilde{\theta_4}$.
• The relations $([y, {}^{y^{-1}}[x,y]] \otimes x)([^{y^{-1}}[x,y], x] \otimes {}^x y)$ and similar expressions map to the identity in $\Gamma_4(P)/\Gamma_4'(P)$, confirming they lie in $\Gamma_4'(P)$.
• Congruences modulo $\Gamma_4'(P)$ allow reduction of complex commutator expressions to trivial elements, validating the closure of the identity set under the required relations.