We prove that the language of all geodesics of any Garside group, with respect to the generating set of divisors of the Garside element, forms a regular language. In particular, the braid groups admit generating sets where the associated language of geodesics is regular.
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… (More)
Recently, right-angled Artin groups have attracted much attention in geometric group theory. They have a rich structure of subgroups and nice algorithmic properties, and they give rise to cubical complexes with a variety of applications. This survey article is meant to introduce readers to these groups and to give an overview of the relevant literature.… (More)
A Garside group is a group admitting a finite lattice generating set D. Using techniques developed by Bestvina for Artin groups of finite type, we construct K(π, 1)s for Garside groups. This construction shows that the (co)homology of any Garside group G is easily computed given the lattice D, and there is a simple sufficient condition that implies G is a… (More)
This paper concerns the homotopy type of hyperplane arrangements associated to infinite Coxeter groups acting as reflection groups on C n. A long-standing conjecture states that the complement of such an arrangement should be aspherical. Some partial results on this conjecture were previously obtained by the author and M. Davis. In this paper, we extend… (More)
Associated to any simplicial graph Γ is a right-angled Artin group AΓ. This class of groups includes free groups and free abelian groups and may be said to interpolate between these two extremes. We study the outer automorphism group of a right-angled Artin group in the case where the defining graph is connected and triangle-free. We give an algebraic… (More)
We study the algebraic structure of the outer automorphism group of a general right-angled Artin group. We show that this group is virtually torsion-free and has finite virtual cohomological dimension. This generalizes results proved in [CCV] for two-dimensional right-angled Artin groups.
T T T T T T T T T T T T T T T Abstract A building is a simplicial complex with a covering by Coxeter complexes (called apartments) satisfying certain combinatorial conditions. A building whose apartments are spherical (respectively Euclidean) Coxeter complexes has a natural piecewise spherical (respectively Euclidean) metric with nice geometric properties.… (More)
We study subgroups and quotients of outer automorphsim groups of right-angled Artin groups (RAAGs). We prove that for all RAAGs, the outer automorphism group is residually finite and, for a large class of RAAGs, it satisfies the Tits alternative. We also investigate which of these automorphism groups contain non-abelian solvable subgroups.