Artin Groups Of Type Research Articles

PALINDROMES AND ORDERINGS IN ARTIN GROUPS

The braid group Bn, endowed with Artin's presentation, admits two distinguished involutions. One is the anti-automorphism rev : Bn→Bn, [Formula: see text], defined by reading braids in the reverse order (from right to left instead of left to right). Another one is the conjugation τ : x ↦ Δ-1xΔ by the generalized half-twist (Garside element).More generally, the involution rev is defined for all Artin groups (equipped with Artin's presentation) and the involution τ is defined for all Artin groups of finite type. A palindrome is an element invariant under rev. We study palindromes and palindromes invariant under τ in Artin groups of finite type. Our main results are the injectivity of the map [Formula: see text] in all finite-type Artin groups, the existence of a left-order compatible with rev for Artin groups of type A, B, D, and the existence of a decomposition for general palindromes. The uniqueness of the latter decomposition requires that the Artin groups carry a left-order.

The $K(\pi,1)$ problem for the affine Artin group of type $\widetilde{B}_n$ and its cohomology

We prove that the complement to the affine complex arrangement of type \widetilde B_n is a K(π,1) space. We also compute the cohomology of the affine Artin group G_{\widetilde B_n} (of type \widetilde B_n ) with coefficients in interesting local systems. In particular, we consider the module ℚ[q^{±1} ,t^{±1}] , where the first n standard generators of G_{\widetilde B_n} act by (−q) -multiplication while the last generator acts by (−t) multiplication. Such a representation generalizes the analogous 1-parameter representation related to the bundle structure over the complement to the discriminant hypersurface, endowed with the monodromy action of the associated Milnor fibre. The cohomology of G_{\widetilde B_n} with trivial coefficients is derived from the previous one.

Uniqueness of roots up to conjugacy for some affine and finite type Artin groups

Let $G$ be one of the Artin groups of finite type ${\mathbf B}_n={\mathbf C}_n$, and affine type $\tilde{\mathbf A}_{n-1}$ and $\tilde{\mathbf C}_{n-1}$. In this paper, we show that if $\alpha$ and $\beta$ are elements of $G$ such that $\alpha^k=\beta^k$ for some nonzero integer $k$, then $\alpha$ and $\beta$ are conjugate in $G$. For the Artin group of type $\mathbf A_n$, this was recently proved by J. Gonz\'alez-Meneses. In fact, we prove a stronger theorem, from which the above result follows easily by using descriptions of those Artin groups as subgroups of the braid group on $n+1$ strands. Let $P$ be a subset of $\{1,...,n\}$. An $n$-braid is said to be \emph{$P$-pure} if its induced permutation fixes each $i\in P$, and \emph{$P$-straight} if it is $P$-pure and it becomes trivial when we delete all the $i$-th strands for $i\not\in P$. Exploiting the Nielsen-Thurston classification of braids, we show that if $\alpha$ and $\beta$ are $P$-pure $n$-braids such that $\alpha^k=\beta^k$ for some nonzero integer $k$, then there exists a $P$-straight $n$-braid $\gamma$ with $\beta=\gamma\alpha\gamma^{-1}$. Moreover, if $1\in P$, the conjugating element $\gamma$ can be chosen to have the first strand algebraically unlinked with the other strands. Especially in case of $P=\{1,...,n\}$, our result implies the uniqueness of root of pure braids, which was known by V. G. Bardakov and by D. Kim and D. Rolfsen.

THE LOWER CENTRAL AND DERIVED SERIES OF THE BRAID GROUPS OF THE FINITELY-PUNCTURED SPHERE

Motivated in part by the study of Fadell–Neuwirth short exact sequences, we determine the lower central and derived series for the braid groups of the finitely-punctured sphere. For n ≥ 1, the class of m-string braid groups Bm(𝕊2\{x1,…,xn}) of the n-punctured sphere includes the usual Artin braid groups Bm(for n = 1), those of the annulus, which are Artin groups of type B (for n = 2), and affine Artin groups of type [Formula: see text] (for n = 3).We first consider the case n = 1. Motivated by the study of almost periodic solutions of algebraic equations with almost periodic coefficients, Gorin and Lin calculated the commutator subgroup of the Artin braid groups. We extend their results, and show that the lower central series (respectively, derived series) of Bmis completely determined for all m ∈ ℕ (respectively, for all m ≠ 4). In the exceptional case m = 4, we obtain some higher elements of the derived series and its quotients.When n ≥ 2, we prove that the lower central series (respectively, derived series) of Bm(𝕊2\{x1,…,xn}) is constant from the commutator subgroup onwards for all m ≥ 3 (respectively, m ≥ 5). The case m = 1 is that of the free group of rank n - 1. The case n = 2 is of particular interest notably when m = 2 also. In this case, the commutator subgroup is a free group of infinite rank. We then go on to show that B2(𝕊2\{x1,x2}) admits various interpretations, as the Baumslag–Solitar group BS(2,2), or as a one-relator group with non-trivial centre for example. We conclude from this latter fact that B2(𝕊2\{x1,x2}) is residually nilpotent, and that from the commutator subgroup onwards, its lower central series coincides with that of the free product ℤ2* ℤ. Further, its lower central series quotients Γi/Γi + 1are direct sums of copies of ℤ2, the number of summands being determined explicitly. In the case m ≥ 3 and n = 2, we obtain a presentation of the derived subgroup, from which we deduce its Abelianization. Finally, in the case n = 3, we obtain partial results for the derived series, and we prove that the lower central series quotients Γi/Γi + 1are 2-elementary finitely-generated groups.

Injectivity on the set of conjugacy classes of some monomorphisms between Artin groups

There are well-known monomorphisms between the Artin groups of finite type $\arA_n$, $\arB_n=\arC_n$ and affine type $\tilde \arA_{n-1}$, $\tilde\arC_{n-1}$. The Artin group $A(\arA_n)$ is isomorphic to the $(n+1)$-strand braid group $B_{n+1}$, and the other three Artin groups are isomorphic to some subgroups of $B_{n+1}$. The inclusions between these subgroups yield monomorphisms $A(\arB_n)\to A(\arA_n)$, $A(\tilde \arA_{n-1})\to A(\arB_n)$ and $A(\tilde \arC_{n-1})\to A(\arB_n)$. There are another type of monomorphisms $A(\arB_d)\to A(\arA_{md-1})$, $A(\arB_d)\to A(\arB_{md})$ and $A(\arB_d)\to A(\arA_{md})$ which are induced by isomorphisms between Artin groups of type $\arB$ and centralizers of periodic braids. In this paper, we show that the monomorphisms $A(\arB_d)\to A(\arA_{md-1})$, $A(\arB_d)\to A(\arB_{md})$ and $A(\arB_d)\to A(\arA_{md})$ induce injective functions on the set of conjugacy classes, and that none of the monomorphisms $A(\arB_n)\to A(\arA_n)$, $A(\tilde \arA_{n-1})\to A(\arB_n)$ and $A(\tilde \arC_{n-1})\to A(\arB_n)$ does so.

Cohomology of affine Artin groups and applications

The result of this paper is the determination of the cohomology of Artin groups of type An, B n and A n with non-trivial local coefficients. The main result is an explicit computation of the cohomology of the Artin group of type B n with coefficients over the module Q[q ±1 ,t ±1 ]. Here the first n - 1 standard generators of the group act by (-q)-multiplication, while the last one acts by (-t)-multiplication. The proof uses some technical results from previous papers plus computations over a suitable spectral sequence. The remaining cases follow from an application of Shapiro's lemma, by considering some well-known inclusions: we obtain the rational cohomology of the Artin group of affine type An as well as the cohomology of the classical braid group Br n with coefficients in the n-dimensional representation presented in Tong, Yang, and Ma (1996). The topological counterpart is the explicit construction of finite CW-complexes endowed with a free action of the Artin groups, which are known to be K(π, 1) spaces in some cases (including finite type groups). Particularly simple formulas for the Euler-characteristic of these orbit spaces are derived.

Conjugacy Problem for Subgroups with Applications to Artin Groups and Braid Type Group

Let G be a group endowed with a solution to the conjugacy problem and with an algorithm which computes the centralizer in G of any element of G. Let H be a subgroup of G. We give some conditions on H, under which we provide a solution to the conjugacy problem in H. We apply our results to some Artin groups and braid type groups. In particular, we give solutions to the explicit conjugacy problem in the Artin groups of type and .

Artin groups of type B and D

Abstract We show that each of the Artin groups of type Bn and Dn can be presented as a semidirect product F ⋊ ℬ n , where F is a free group and ℬ n is the n-string braid group. We explain how these semidirect product structures arise quite naturally from fibrations, and observe that, in each case, the action of the braid group ℬ n on the free group F is classical. We prove that, for each of the semidirect products, the group of automorphisms which leave invariant the normal subgroup F is small: namely, Out(A(Bn ), F ) has order 2, and Out(A(Dn ), F ) has order 4 if n is even and 2 if n is odd. It is known that the Artin group of type Dn may be viewed as an index 2 subgroup of the n-string braid group over a disk with a degree 2 orbifold point. We show that this orbifold braid group has outer automorphism group of order 2, for all n ≥ 2.

BMW algebras of simply laced type

It is known that the recently discovered representations of the Artin groups of type A n , the braid groups, can be constructed via BMW algebras. We introduce similar algebras of type D n and E n which also lead to the newly found faithful representations of the Artin groups of the corresponding types. We establish finite dimensionality of these algebras. Moreover, they have ideals I 1 and I 2 with I 2 ⊂ I 1 such that the quotient with respect to I 1 is the Hecke algebra and I 1 / I 2 is a module for the corresponding Artin group generalizing the Lawrence–Krammer representation. Finally we give conjectures on the structure, the dimension and parabolic subalgebras of the BMW algebra, as well as on a generalization of deformations to Brauer algebras for simply laced spherical type other than A n .

Conjugacy problem for braid groups and Garside groups

We present a new algorithm to solve the conjugacy problem in Artin braid groups, which is faster than the one presented by Birman, Ko, and Lee [Adv. Math. 139 (1998) 322–353]. This algorithm can be applied not only to braid groups, but to all Garside groups (which include finite type Artin groups and torus knot groups among others).

Braid pictures for Artin groups

We define the braid groups of a two-dimensional orbifold and introduce conventions for drawing braid pictures. We use these to realize the Artin groups associated to the spherical Coxeter diagrams A n , B n = C n and D n and the affine diagrams A n , B n , C n and D n as subgroups of the braid groups of various simple orbifolds. The cases D n , B n , C n and D n are new. In each case the Artin group is a normal subgroup with abelian quotient; in all cases except A n the quotient is finite. We also illustrate the value of our braid calculus by giving a picture-proof of the basic properties of the Garside element of an Artin group of type D n .

THE TITS CONJECTURE FOR LOCALLY REDUCIBLE ARTIN GROUPS

Given an Artin system (A,S), a conjecture of Tits states that the subgroup A(2) of A generated by the squares of the generators in S is subject only to the obvious commutator relations between generators. In particular, A(2) is a right-angled Artin group. We prove this conjecture for a class of infinite type Artin groups, called locally reducible Artin groups, for which the associated Deligne complex has a CAT(0) geometry. We also prove that for any special subgroup AT of A, A(2)∩AT=(AT)(2).

On completeness of word reversing

Word reversing is a combinatorial operation on words that detects pairs of equivalent words in monoids that admit a presentation of a certain form. Here we give conditions for this method to be complete in the sense that every pair of equivalent words can be detected by word reversing. In addition, we obtain explicit upper bounds on the complexity of the process. As an application, we show that Artin groups of Coxeter type B embed into Artin groups of type A and are left orderable.

Symmetrical Subgroups of Artin Groups

We show that the subgroup fixed by a group of symmetries of an Artin system (A, S) is itself an Artin group under the hypothesis that the Deligne complex associated to A admits a suitable CAT(0) metric. Such a metric is known to exist for all Artin groups of type FC, which include all the finite type Artin groups as well as many infinite types. We also recover the previously known analogous result for an arbitrary Coxeter system (W, S).

A Geometric Rational Form for Artin Groups of FC Type

If A is an Artin group whose poset of finite type special subgroups is a flag complex, then A is said to be of FC type. Such groups act cocompactly on a CAT(0) cubical complex with finite type Artin groups as stabilizers. We use the geometry of this complex to obtain a rational normal form for the group.

Gaussian Groups are Torsion Free

Assume thatGis a group of fractions of a cancellative monoid where lower common multiples exist and divisibility has no infinite descending chain. ThenGis torsion free. The result applies in particular to all finite Coxeter type Artin groups.

Automorphism groups of some affine and finite type Artin groups

We observe that, for fixed n ≥ 3, each of the Artin groups of finite type An, Bn = Cn, and affine type ˜ An−1 and ˜ Cn−1 is a central extension of a finite index subgroup of the mapping class group of the (n + 2)-punctured sphere. (The centre is trivial in the affine case and infinite cyclic in the finite type cases). Using results of Ivanov and Korkmaz on abstract commensurators of surface mapping class groups we are able to determine the automorphism groups of each member of these four infinite families of Artin groups. A rank n Coxeter matrix is a symmetric n × n matrix M with integer entries mij ∈ N ∪ {∞} where mij ≥ 2 for ij, and mii = 1 for all 1 ≤ i ≤ n. Given any rank n Coxeter matrix M, the Artin group of type M is defined by the presentation

The Homotopy Type of Artin Groups

A b st ract . Let W be a group generated by reflections in Rn. W acts on the complement Y ⊂ Cn of the complexification of the reflection hyperplanes of W. The fundamental group of the orbit space Y/W is the so called Artin group of type W. Here we give a new description of the homotopy type of Y/W in terms of a convex polyhedrum in Rn with identifications on the faces. Such identifications are quite easy to describe and are naturally connected to the combinatorics of W. We derive an associated algebraic complex which computes the cohomology of local systems on Y/W: its kth-module is freely generated by the k-subsets of {1, . . . , n} and the coboundary is explicitly given by a formula involving the Poincare series of the group. In particular, we are able to compute the cohomology of the Artin group associated to W for all the exceptional groups.