In this paper we study some combinatorial aspects of the singular Artin monoids. Firstly, we show that a singular Artin monoid SA can be presented as a semidirect product of a graph monoid with its associated Artin group A. Such a decomposition implies that a singular Artin monoid embeds in a group. Secondly, we give a solution to the word problem for the FC type singular Artin monoids. Afterwa...

We interpret the Artin–Rees lemma and the Izumi theorem in term of Artin function and we obtain a stable version of the Artin–Rees lemma. We present different applications of these interpretations. First we show that the Artin function of X1X2 −X3X4, as a polynomial in the ring of power series in more than three variables, is not bounded by an affine function. Then we prove that the Artin funct...

Let (W, S) be a Coxeter system, with W a finite, irreducible Coxeter group and let GW be the associated Artin group (see Bourbaki [5] for an introduction to Coxeter groups and their classifications and Brieskorn and Saito [6] for relations between Coxeter groups and Artin groups ). The main objects of study of this paper are the Artin groups of type An . We recall that the Artin group GAn is th...

1 0.1. Braid groups 1 0.2. Artin–Tits groups 2 1. The general case 2 1.1. The word reversing technique 3 1.2. Artin–Tits monoids 4 1.3. Artin–Tits groups 5 1.4. Exercises 5 2. The spherical case 6 2.1. Background about Coxeter groups 6 2.2. Garside structure 7 2.3. Normal form 7 2.4. Exercises 8 3. The braid case 9 3.1. The Artin representation 10 3.2. Handle reduction 11 3.3. The braid orderin...

trace function 45, 46-48, 50, 51, 56, 62, 64 adapted stratification 12, 24, 26, 37, 38, 46, 48, 63, 121, 408 additive character sums 2, 7, 8, 111-158, 161-183, 270-280, 427-428 analytic rank 8, 443, 444, 445, 447, 449, 451, 453 analytic rank, geometric see "geometric analytic rank" analytic rank, quadratic see "quadratic analytic rank" Analytic Rank Theorem 445, 450-454 approximate trace fu...

We show the existence of a first order theory Cmd,e whose Noetherian models are precisely the local Cohen-Macaulay rings of dimension d and multiplicity e. The completion of a model of Cmd,e is again a model and is moreover Noetherian. If R is an equicharacteristic local Gorenstein ring of dimension d and multiplicity e with algebraically closed residue field and if the Artin Approximation Prop...

The Coxeter groups that act geometrically on euclidean space have long been classified and presentations for the irreducible ones are encoded in the well-known extended Dynkin diagrams. The corresponding Artin groups are called euclidean Artin groups and, despite what one might naively expect, most of them have remained fundamentally mysterious for more than forty years. Recently, my coauthors ...

We survey the role of right-angled Artin groups in the theory of diffeomorphism groups of low dimensional manifolds. We first describe some of the subgroup structure of right-angled Artin groups. We then discuss the interplay between algebraic structure, compactness, and regularity for group actions on one–dimensional manifolds. For compact one–manifolds, every right-angled Artin group acts fai...

Introduction. Witt [5] proved that two binary or ternary quadratic forms, over an arbitrary field (of characteristic not 2) are equivalent if and only if they have the same determinant and Hasse invariant. His proof is brief and elegant but uses a lot of the theory of simple algebras. The purpose of this note is to make this fundamental theorem more accessible by giving a short proof using only...

In this paper we study the growth rates of Artin monoids and we show that 4 is a universal upper bound. We also show that the generating functions of the associated right-angled Artin monoids are given by families of Chebyshev polynomials. Applications to Artin groups and positive braids are given.