Luis Paris

Gaussian Groups and Garside Groups, Two Generalisations of Artin Groups

It is known that a number of algebraic properties of the braid groups extend to arbitrary finite Coxeter-type Artin groups. Here we show how to extend the results to more general groups that we call Garside groups. Define a Gaussian monoid to be a finitely generated cancellative monoid where the expressions of a given element have bounded lengths, and where left and right lowest common multiples exist. A Garside monoid is a Gaussian monoid in which the left and right lowest common multiples satisfy an additional symmetry condition. A Gaussian group is the group of fractions of a Gaussian monoid, and a Garside group is the group of fractions of a Garside monoid. Braid groups and, more generally, finite Coxeter-type Artin groups are Garside groups. We determine algorithmic criteria in terms of presentations for recognizing Gaussian and Garside monoids and groups, and exhibit infinite families of such groups. We describe simple algorithms that solve the word problem in a Gaussian group, show that these algorithms have a quadratic complexity if the group is a Garside group, and prove that Garside groups have quadratic isoperimetric inequalities. We construct normal forms for Gaussian groups, and prove that, in the case of a Garside group, the language of normal forms is regular, symmetric, and geodesic, has the 5-fellow traveller property, and has the uniqueness property. This shows in particular that Garside groups are geodesically fully biautomatic. Finally, we consider an automorphism of a finite Coxeter-type Artin group derived from an automorphism of its defining Coxeter graph, and prove that the subgroup of elements fixed by this automorphism is also a finite Coxeter-type Artin group that can be explicitly determined. 1991 Mathematics Subject Classification: primary 20F05, 20F36; secondary 20B40, 20M05.

The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group

Abstract.Let A be an Artin group with standard generating set {σs:s∈S}. Tits conjectured that the only relations in A amongst the squares of the generators are consequences of the obvious ones, namely that σs2 and σt2 commute whenever σs and σt commute, for s,t∈S. In this paper we prove Tits’ conjecture for all Artin groups. In fact, given a number ms≥2 for each s∈S, we show that the elements {Ts=σsms:s∈S} generate a subgroup that has a finite presentation in which the only defining relations are that Ts and Tt commute if σs and σt commute.

Presentations for the punctured mapping class groups in terms of Artin groups

Consider an oriented compact surface F of positive genus, possibly with boundary, and a finite set P of punctures in the interior of F, and define the punctured mapping class group of F relatively to P to be the group of isotopy classes of orientation-preserving homeomorphisms h : F → F which pointwise fix the boundary of F and such that h(P) = P. In this paper, we calculate presentations for all punctured mapping class groups. More precisely, we show that these groups are isomorphic with quotients of Artin groups by some relations involving fundamental elements of parabolic subgroups.

Vassiliev invariants for braids on surfaces

We show that Vassiliev invariants separate braids on a closed oriented surface, and we exhibit a universal Vassiliev invariant for these braids in terms of chord diagrams labeled by elements of the fundamental group of the surface.

IRREDUCIBLE COXETER GROUPS

We prove that a non-spherical irreducible Coxeter group is (directly) indecomposable and that an indefinite irreducible Coxeter group is strongly indecomposable in the sense that all its finite index subgroups are (directly) indecomposable. Let W be a Coxeter group. Write W = WX1 × ⋯ × WXb × WZ3, where WX1, … , WXb are non-spherical irreducible Coxeter groups and WZ3 is a finite one. By a classical result, known as the Krull–Remak–Schmidt theorem, the group WZ3 has a decomposition WZ3 = H1 × ⋯ × Hq as a direct product of indecomposable groups, which is unique up to a central automorphism and a permutation of the factors. Now, W = WX1 × ⋯ × WXb × H1 × ⋯ × Hq is a decomposition of W as a direct product of indecomposable subgroups. We prove that such a decomposition is unique up to a central automorphism and a permutation of the factors. Write W = WX1 × ⋯ × WXa × WZ2 × WZ3, where WX1, … , WXa are indefinite irreducible Coxeter groups, WZ2 is an affine Coxeter group whose irreducible components are all infinite, and WZ3 is a finite Coxeter group. The group WZ2 contains a finite index subgroup R isomorphic to ℤd, where d = |Z2| - b + a and b - a is the number of irreducible components of WZ2. Choose d copies R1, … , Rd of ℤ such that R = R1 × ⋯ × Rd. Then G = WX1 × ⋯ × WXa × R1 × ⋯ × Rd is a virtual decomposition of W as a direct product of strongly indecomposable subgroups. We prove that such a virtual decomposition is unique up to commensurability and a permutation of the factors.

The proof of Birman’s conjecture on singular braid monoids

Let B_n be the Artin braid group on n strings with standard generators sigma_1, ..., sigma_{n-1}, and let SB_n be the singular braid monoid with generators sigma_1^{+-1}, ..., sigma_{n-1}^{+-1}, tau_1, ..., tau_{n-1}. The desingularization map is the multiplicative homomorphism eta: SB_n --> Z[B_n] defined by eta(sigma_i^{+-1}) =_i^{+-1} and eta(tau_i) = sigma_i - sigma_i^{-1}, for 1 <= i <= n-1. The purpose of the present paper is to prove Birmans conjecture, namely, that the desingularization map eta is injective.

Residual "p" properties of mapping class and surface groups

Let M(Σ, P) be the mapping class group of a punctured oriented surface (E, P) (where P may be empty), and let T p (Σ, P) be the kernel of the action of M(Σ,P) on H 1 (Σ \ P,F p ). We prove that T p (Σ,P) is residually p. In particular, this shows that M(Σ,P) is virtually residually p. For a group G we denote by Ip(G) the kernel of the natural action of Out(G) on H 1 (G,F p ). In order to achieve our theorem, we prove that, under certain conditions (G is conjugacy p-separable and has Property A), the group Ip(G) is residually p. The fact that free groups and surface groups have Property A is due to Grossman. The fact that free groups are conjugacy p-separable is due to Lyndon and Schupp. The fact that surface groups are conjugacy p-separable is, from a technical point of view, the main result of the paper.

Universal cover of Salvetti's complex and topology of simplicial arrangements of hyperplanes

Let V be a real vector space. An arrangement of hyperplanes in V is a finite set A of hyperplanes through the origin. A chamber of A is a connected component of V − (∪ H ∈ A H). The arrangement A is called simplicial if ∩ H ∈ A H = {0} and every chamber of A is a simplicial cone. For an arrangement A of hyperplanes in V, we set ... (formule)... where V C = C ⊗ V is the complexification of V, and, for H ∈ A, H C is the complex hyperplane of V C spanned by H. Let A be an arrangement of hyperplanes of V. Salvetti constructed a simplicial complex Sal(A) and proved that Sal(A) has the same homotopy type as M(A). In this paper we give a new short proof of this fact

Combinatorics of inductively factored arrangements

Abstract A real arrangement of hyperplanes is a finite family A of hyperplanes through the origin in a finite-dimensional real vector space V = R1. A real arrangement A of hyperplanes is said to be factored if there exists a partition Π = (Π1, …, Π1) of A into l disjoint subsets such that the Orlik-Solomon algebra of A factors according to this partition. A real arrangement A of hyperplanes is called inductively factored if it is factored and there exists a hyperplane H ϵ A such that the arrangement obtained by removing H from A and the arrangement on H consisting of all intersections of elements of A — H with H are both inductively factored. A chamber of A is a connected component of the complement of A . For a fixed base chamber, we may define a partial order on the set of chambers according to their combinatorial distances from the base chamber. Given an inductive factorization Π = (Π1,…(Π1 and a base chamber C0, we define the counting map of A with respect to C0 as a morphism from the poset of chambers to the poset 0, 1,…,|Π1| x·.x (0, 1,…,Π1). We prove that, for a suitable base chamber, the counting map is a bijection, the poset of chambers is a lattice, and its rank-generating function has a nice factorization. We consider the dual decomposition of the sphere of V induced by A . We prove that, if A is inductively factored, then this cellular decomposition can be viewed as a decomposition of the boundary of the l-cube [0,|Π1|]x·.x]0, |Π1|[ by cubic cells.

A note on the Lawrence-Krammer-Bigelow representation

A very popular problem on braid groups has recently been solved by Bigelow and Krammer, namely, they have found a faithful linear representation for the braid group Bn. In their papers, Bigelow and Kram- mer suggested that their representation is the monodromy representation of a certain fibration. Our goal in this paper is to understand this mon- odromy representation using standard tools from the theory of hyperplane arrangements. In particular, we prove that the representation of Bigelow and Krammer is a sub-representation of the monodromy representation which we consider, but that it cannot be the whole representation. AMS Classification 20F36; 52C35, 52C30, 32S22