We construct a quasi-Garside monoid structure for the free group. This monoid should be thought of as a dual braid monoid for the free group, generalising the constructions by Birman-Ko-Lee and by the author of new Garside monoids for Artin groups of spherical type. Conjecturally, an analog construction should be available for arbitrary Artin groups and for braid groups of well-generated comple...

Abstract. We define an Artin prime for an integer g to be a prime such that g is a primitive root modulo that prime. Let g ∈ Z \ {−1} and not be a perfect square. A conjecture of Artin states that the set of Artin primes for g has a positive density. In this paper we study a generalization of this conjecture for the primes produced by a polynomial and explore its connection with the problem of ...

We prove that every separated Artin stack of finite type over a noetherian base scheme admits a proper covering by a quasi–projective scheme. An application of this result is a version of the Grothendieck existence theorem for Artin stacks.

In this paper we will present the results of Artin– Markov on braid groups by using the Gröbner-Shirshov basis. As a consequence we can reobtain the normal form of Artin–Markov– Ivanovsky as an easy corollary.

In this paper we consider the class of 2–dimensional Artin groups with connected, large type, triangle-free defining graphs (type CLTTF). We classify these groups up to isomorphism, and describe a generating set for the automorphism group of each such Artin group. In the case where the defining graph has no separating edge or vertex we show that the Artin group is not abstractly commensurable t...

The action dimension of a discrete group G is the minimum dimension of a contractible manifold, which admits a proper G-action. In this dissertation, we study the action dimension of general Artin groups. The main result is that the action dimension of an Artin group with the nerve L of dimension n for n 6= 2 is less than or equal to (2n+1) if the Artin group satisfies the K(π, 1)-Conjecture an...

Artin groups span a wide range of groups from braid groups to free groups to free abelian groups, as well as many more exotic groups. In recent years, Artin groups and their subgroups have proved to be a rich source of examples and counterexamples of interesting phenomena in geometry and group theory. Artin groups, like Coxeter groups, are defined by presentations of a particular form. A Coxete...