We show that the simple elements of the dual Garside structure of an Artin group of type Dn are Mikado braids, giving a positive answer to a conjecture of Digne and the second author. To this end, we use an embedding of the Artin group of type Dn in a suitable quotient of an Artin group of type Bn noticed by Allcock, of which we give a simple algebraic proof here. This allows one to give a char...

We compute the relative divergence and the subgroup distortion of Bestvina-Brady subgroups. We also show that for each integer n ≥ 3, there is a free subgroup of rank n of some right-angled Artin group whose inclusion is not a quasi-isometric embedding. This result answers the question of Carr about the minimum rank n such that some rightangled Artin group has a free subgroup of rank n whose in...

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 An, Bn = Cn and Dn and the affine diagrams Ãn, B̃n, C̃n and D̃n as subgroups of the braid groups of various simple orbifolds. The cases Dn, B̃n, C̃n and D̃n are new. In each case the Artin group is a normal...

It is shown that the compressed word problem of a graph product of finitely generated groups is polynomial time Turing-reducible to the compressed word problems of the vertex groups. A direct corollary of this result is that the word problem for the automorphism group of a right-angled Artin group or a right-angled Coxeter group can be solved in polynomial time. Moreover, it is shown that a res...

This paper studies the homological determinants and Nakayama automorphisms of not-necessarily-noetherian m-Koszul twisted Calabi-Yau or, equivalently, m-Koszul Artin-Schelter regular, algebras. Dubois-Violette showed that such an algebra is isomorphic to a derivation quotient algebra D(w, i) for a unique-up-to-scalar-multiples twisted superpotential w. By definition, D(w, i) is the quotient of ...

Consider the mapping class group Modg,p of a surface Σg,p of genus g with p punctures, and a finite collection {f1, . . . , fk} of mapping classes, each of which is either a Dehn twist about a simple closed curve or a pseudo-Anosov homeomorphism supported on a connected subsurface. In this paper we prove that for all sufficiently large N , the mapping classes {f 1 , . . . , f k } generate a rig...

1. Basics 2 1.1. Commutants 2 1.2. Opposite Rings 3 1.3. Units 3 1.4. Ideals 4 1.5. Modules 4 1.6. Division rings 5 1.7. Endomorphism rings 7 1.8. Monoid Rings 8 1.9. Noetherian and Artinian Rings 11 1.10. Simple rings 15 1.11. Prime ideals and prime rings 17 1.12. Notes 18 2. Wedderburn-Artin Theory 18 2.1. Semisimple modules and rings 18 2.2. Wedderburn-Artin I: Semisimplicity of Mn(D) 20 2.3...