A certain structure of Artin groups and the isomorphism conjecture

Abstract

We observe an inductive structure in a large class of Artin groups and exploit this information to deduce the Farrell-Jones isomorphism conjecture for several classes of Artin groups of finite real, complex and affine types.As an application, in Theorem 3.4, we extend our earlier computation of the surgery groups of the pure braid groups in [29], to the finite type pure Artin groups appearing in Theorem 1.1.Corollary 1.1. For any subgroup of Γ, where Γ is as in Theorem 1.1 the two assembly maps A K and A L are isomorphisms.Below we mention a well known consequence of the isomorphism of the K-theory assembly map. See [[12], §1.6.1].Corollary 1.2. The Whitehead group W h(−), the reduced projective class group K0 (−) and the negative K-groups K −i (−), for i ≥ 1, vanish for any subgroup of the groups considered in Theorem 1.1.The paper is organized as follows. In Section 2 we recall some background on Artin groups and in Section 3 we state our results which are used to prove the main theorem. We describe orbifold braid groups and some work from [1] in Section 4, which is the key input to this work. Section 5 contains the proofs of the results of Section 3. The computation of the surgery groups of pure Artin groups is given in Section 6. Finally, in Section 7 we recall the statement of the isomorphism conjecture and all the basic facts needed to prove Theorem 3.3.We end the Introduction with the following remark.

The Farrell-Jones Conjecture for normally poly-free groups

“…Conjecture for all Artin groups of FC-type [13]. Sayed K. Roushon has also proved the conjecture for some classes of Artin groups [19].…”

Poly-freeness of Artin groups and the Farrell-Jones Conjecture

“…Here by Farrell-Jones Conjecture, we mean the Farrell-Jones Conjecture with finite wreath products and coefficients in additive categories (which we will abbreviate to FJCw), see [9,19,22] and the references therein for more information. Many Artin groups are known to satisfy FJCw, for example Artin groups of XXL type [18, Corollary E], Artin groups of FC-type [16] and some classes of affine Artin groups [23]. There are also many results regarding Artin groups which have similar flavor, see for example [12,14,15].…”