A Geometric and Algebraic Description of Annular Braid Groups

Abstract

We provide a new presentation for the annular braid group. The annular braid group is known to be isomorphic to the finite type Artin group with Coxeter graph Bn. Using our presentation, we show that the annular braid group is a semidirect product of an infinite cyclic group and the affine Artin group with Coxeter graph Ãn - 1. This provides a new example of an infinite type Artin group which injects into a finite type Artin group. In fact, we show that the affine braid group with Coxeter graph Ãn - 1 injects into the braid group on n + 1 stings. Recently it has been shown that the braid groups are linear, see [3]. Therefore, this shows that the affine braid groups are also linear.

The braid group of a necklace

“…topologically an annulus). The circular braid group C B n admits the following presentation (where the indices are defined modulo n, see for instance [16]):…”

“…By comparison of group presentations one deduces that C B n =Ã n−1 ζ (see also [7,16]). It then follows from Theorem 2.3 and Remark 2.5 that π 1 (Conf L n ) =Ã n−1 τ .…”

Injections of Artin groups

“…A complete description of these isomorphisms appears in the paper of Charney-Crisp [9]. The proofs are due to Allcock, Kent-Peifer, Charney-Peifer, Crisp, and Charney-Crisp ( [1], [25], [10], [12], [9]). [9]).…”

Homology computations for complex braid groups

“…We can give a presentation for A Br that is different from the one provided before (see [KP02]). We define τ = σ 1 σ 2 · · · σ r and σ 1 = τ −1 σ 2 τ .…”

“…According to [KP02] and [BMR98] the center of B(de, e, r) is generated by β(de, e, r) = (τ e ) (r/r∧e) . Hence it follows that in the quotient B(de, e, r)/Z(B(de, e, r)) there is an element, namely (τ e ), that has order at most (r/r ∧ e) and is the image of a root of the generator of the center of B(de, e, r).…”