Topology Atlas | Conferences

Action of braid groups related to double branched covers
by
Jozef H. Przytycki
GWU
Coauthors: Mieczyslaw Dabkowski (GWU)

There is a classical result that the Burau representation of the 3-braid group reduces at t = −1 to the representation φ₁: B₃ → SL(2, Z) with the kernel generated by (σ₁σ₂)⁶ (that is the square of the center of B₃). The Burau representation at t = −1 is know to be related to the action of a braid group on the homology of the double branch cover of a punctured disk. We consider here the generalization of the above construction to the action of a braid group on the graded Lie ring associated to the lower central series of the 2-generator free group. The lower central series of a group G (G₁ = G, G₂ = [G, G], ..., G_n = [G_n−1, G]) yields the associated graded Lie ring of the group: L = L₁ ⊕ L₂ ⊕ ... ⊕ L_i ⊕ ... where L_i = G_i/G_i+1. The Lie bracket in L corresponds to the group bracket [g, h] = g⁻¹h⁻¹gh. We computed the representation φ_i: B₃ → aut(L_i) for i ≤ 5 and in every case ker(φ_i) contains (σ₁σ₂)⁶. For example φ₅ → SL(6, Z) is given by: φ₅(σ₁) =

é ê ê ê ê ê ê ê ê ê ë 1 1 1 1 1 1 0 1 2 0 1 2 0 0 1 0 0 1 0 0 0 1 1 1 0 0 0 0 1 2 0 0 0 0 0 1 ù ú ú ú ú ú ú ú ú ú û

and φ₅(σ₂) =  

é ê ê ê ê ê ê ê ê ê ë 1 0 0 0 0 0 2 1 0 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 2 1 0 2 1 0 1 1 1 1 1 1 ù ú ú ú ú ú ú ú ú ú û

We speculate on the usefulness of these representations and compare to actions of B₃ on Burnside groups, in particular to the fact that (σ₁σ₂)⁶ acts non-trivially on G/G₄ for G being the Burnside group on two generators and exponent five.

Date received: January 7, 2003

Copyright © 2003 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cajr-32.