In this article we construct a piecewise Euclidean, non-positively curved 2-complex for the 3-generator Artin groups of large type. As a consequence we show that these groups are biautomatic. A slight modification of the proof shows that many other Artin groups are also biautomatic. The general question (whether all Artin groups are biautomatic) remains open.

In this paper, we survey some recent results on the Artin conjecture and discuss some aspects for the Artin conjecture.

The Tits Conjecture, proved by Crisp and Paris, states that squares of the standard generators any Artin group generate an obvious right-angled subgroup. We consider a larger set elements consisting all centers irreducible spherical special subgroups group, conjecture sufficiently large powers those This alleged subgroup is in some sense as possible; its nerve homeomorphic to ambient group. ver...

We conjecture that the word problem of Artin–Tits groups can be solved without introducing trivial factors ss or ss. Here we make this statement precise and explain how it can be seen as a weak form of hyperbolicity. We prove the conjecture in the case of Artin–Tits groups of type FC, and we discuss various possible approaches for further extensions, in particular a syntactic argument that work...

We study certain quotients of generalized Artin groups which have a natural map onto D-type Artin groups, where the generalized Artin group A(T ) is defined by a signed graph T . Then we find a certain quotient G(T ) according to the graph T , which also have a natural map onto A(Dn). We prove that G(T ) is isomorphic to a semidirect product of a group K(m,n), with the Artin group A(Dn), where ...

We prove by explicit construction that graph braid groups and most surface groups can be embedded in a natural way in right-angled Artin groups, and we point out some consequences of these embedding results. We also show that every right-angled Artin group can be embedded in a pure surface braid group. On the other hand, by generalising to rightangled Artin groups a result of Lyndon for free gr...

In the present work, we give necessary and sufficient conditions on the graph of a right-angled Artin group that determine whether the group is subgroup separable or not. Also, we show that right-angled Artin groups are conjugacy separable. Moreover, we investigate the profinite topology of F 2 × F 2 and of the group L in [22], which are the only obstructions for the subgroup separability of th...

In this paper we propose right-angled Artin groups as platform for a secret sharing scheme based on the efficiency (linear time) of the word problem. We define two new problems: subgroup isomorphism problem for Artin subgroups and group homomorphism problem in right-angled Artin groups. We show that the group homomorphism and graph homomorphism problems are equivalent, and the later is known to...

For each Artin group we compute the reduced `cohomology of (the universal cover of) its “Salvetti complex”. This is a CW-complex which is conjectured to be a model for the classifying space of the Artin group. In the many cases when this conjecture is known to hold our calculation describes the reduced `-cohomology of the Artin group.

In this paper, we consider the kernel of the natural projection from the Artin group of dihedral type I2(k) to the corresponding Coxeter group, which we call the pure Artin group of dihedral type. We present a rational function expression for the spherical growth series of the pure Artin group of dihedral type with respect to a natural generating set.