”Braidiness” of type D_{n} Artin groups (La ”tressitude” de A(D_{n}))

John Crisp – joint work with Luis Paris

It is well-known that the Artin group of type A(B_{n}) decomposes as a semi-direct product with quotient the n-sting braid group (type A_{n−1} Artin group) and kernel free of rank n.

A(B_{n}) ≅ F_{n} ⋊_{ρ_{B}} B_{n}

The conjugation action of B_{n} on F_{n} in this product is determined by the famous Artin representation which we denote ρ_{B} : B_{n} → Aut(F_{n}) and which was shown by Artin to be faithful. This decomposition can be explained in terms of a fibring of the associated hyperplane complement of type B_{n} and also in terms of braid pictures.

It is a lesser known fact that there is also a decomposition of the Artin group of type D_{n} as a semi-direct product:

A(D_{n}) ≅ F_{n−1} ⋊_{ρ_{D}} B_{n}

We explain how this can be derived from a fibring of the hyperplane complement of type D_{n} (closely related to that of type B_{n}) and also, as with the previous case, how this may be interpreted by braid pictures (braid pictures à la D. Allcock). The fibration one considers is of a ramified quotient of the hyperplane complement where the fibre is a complex plane with n singular points of degree 2 and has fundamental group K = ⋆_{n}C_{2}. Thus we construct a semi-direct product K ⋊ B_{n} which may also be seen as the braid group over the orbifold “a disk with one singular point of degree 2”. The group A(D_{n}) is the index 2 subgroup of this semidirect product obtained by restricting the kernel to the characteristic free subgroup F_{n−1} = ker(K → C_{2}).

We present certain properties of the “new” representation ρ_{D} : B_{n} → Aut(F_{n−1}):

(i) It differs qualitatively from Artin’s representation ρ_{B} in that the pure braid group acts nontrivially “on homology” (on the abelianisation Z^{n−1} of F_{n−1}) whereas the abelianisation of ρ_{B} has the pure braid group for its kernel.

(ii) ρ_{D} is faithful.

(iii) There are no nontrivial automorphism of F_{n−1} which are equivariant with respect to ρ_{D}. (This is motivated by the problem “Is Out(A(D_{n})) finite?”, and inspired by the proof in Dyer-Grossman for the representation ρ_{B}. They actually show that Out(A(A_{n−1})) has order 2).

We finish with a construction of a faithful “Artin-Magnus type” representation ρ of the braid group by automorphisms of G = ⋆_{n}H (given any group H and a distinguished element δ ∈ H). If H is a Garside group with fundamental element δ then the semi-direct product G ⋊_{ρ} B_{n} is a Garside group. Post Scriptum: Shortly after presenting this talk I realised that the representation ρ_{D} may be obtained as follows. Following Perron-Vannier, the braid group B_{n} may be represented faithfully by sending each standard generator to a Dehn twist in a certain surface S with boundary. (More precisely the surface is a “daisychain” of n − 1 annuli each glued to the next on a square patch – the Dehn twists are along the centrelines of the annuli). The representation ρ_{D} is just that given by the action of B_{n} on π_{1}(S) = F_{n−1} !! The “action on homology” is just that, and gives a representation of the braid group B_{n} in SL(n − 1, Z) which is actually a specialisation of the Burau representation. (Its kernel is obviously related to the Torelli subgroup of the mapping class group of S.) Finally, there is a “hyperelliptic” involution of S whoes quotient is the orbifold with n singular points of degree 2. Thus one can also recover from this picture the braid group action on the larger group ⋆_{n}C_{2}.