Groupes de Garside

Define a Garside monoid to be a cancellative monoid where right and left lcm's exist and that satisfy additional finiteness assumptions, and a Garside group to be the group of fractions of a Garside monoid. The family of Garside groups contains the braid groups, all spherical Artin-Tits groups, and various generalizations previously considered.² Here we prove that Garside groups are biautomatic, and that being a Garside group is a recursively enumerable property, i.e., there exists an algorithm constructing the (infinite) list of all small Gaussian groups. The latter result relies on an effective, tractable method for recognizing those presentations that define a Garside monoid.     

THE CONJUGACY PROBLEM IN SMALL GAUSSIAN GROUPS

Small Gaussian groups are a natural generalization of spherical Artin groups, namely groups of fractions of monoids in which the existence of least common multiples is kept as an hypothesis, but the relations between the generators are not supposed to necessarily be of Coxeter type. We show here how to extend the Elrifai–Morton solution for the conjugacy problem in braid groups to every small Gaussian group.     

Bestvina's Normal Form Complex and the Homology of Garside Groups

A Garside group is a group admitting a finite lattice generating set . 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 , and there is a simple sufficient condition that implies G is a duality group. The universal covers of these K(π, 1)s enjoy Bestvina's weak nonpositive curvature condition. Under a certain tameness condition, this implies that every solvable subgroup of G is virtually Abelian.     

Characterizing Automorphism Groups of Ordered Abelian Groups

A proof is given of the following theorem, which characterizesfull automorphism groups of ordered abelian groups: a groupH is the automorphism group of some ordered abelian group ifand only if H is right-orderable. 2000 Mathematics Subject Classification20K15, 20K20, 20F60, 20K30 (primary); 03E05 (secondary).     

On the inverse braid monoid

Inverse braid monoid describes a structure on braids where the number of strings is not fixed. So, some strings of initial n may be deleted. In the paper we show that many properties and objects based on braid groups may be extended to the inverse braid monoids. Namely we prove an inclusion into a monoid of partial monomorphisms of a free group. This gives a solution of the word problem. Another solution is obtained by an approach similar to that of Garside. We give also the analogues of Artin presentation with two generators and Sergiescu graph-presentations.     

Group Laws and Free Subgroups in Topological Groups

A proof is given that a permutation group in which differentfinite sets have different stabilizers cannot satisfy any grouplaw. For locally compact topological groups with this property,almost all finite subsets of the group are shown to generatefree subgroups. Consequences of these theorems are derived for:Thompson's group F, weakly branch groups, automorphism groupsof regular trees, and profinite groups with alternating compositionfactors of unbounded degree. 2000 Mathematics Subject Classification20B07, 20E10, 20E18, 20P05.     

Bounded Rank Subgroups of Coxeter Groups, Artin Groups and One-Relator Groups with Torsion

We obtain a number of results regarding the freeness of subgroupsof Coxeter groups, Artin groups and one-relator groups withtorsion. In the case of Coxeter groups, we also obtain resultson quasiconvexity and subgroup separability. 2000 MathematicsSubject Classification 20F65, 20F55, 20F36, 20F06.     

Artin monoids inject in their groups

We prove that the natural homomorphism from an Artin monoid to its associated Artin group is always injective. Received: March 14, 2002     

The dual braid monoid

We study a new monoid structure for Artin groups associated with finite Coxeter systems. Like the classical positive braid monoid, the new monoid is a Garside monoid. We give several equivalent constructions: algebraically, the new monoid arises when studying Coxeter systems in a “dual” way, replacing the pair (W,S) by (W,T), with T the set of all reflections; geometrically, it arises when looking at the reflection arrangement from a certain basepoint. In the type A case, we recover the monoid constructed by Birman, Ko and Lee.     

Polynomial Dehn Functions and the Length of Asynchronously Automatic Structures

We extend the range of observed behaviour among length functionsof optimal asynchronously automatic structures. We do so bymeans of a construction that yields asynchronously automaticgroups with finite aspherical presentations where the Dehn functionof the group is polynomial of arbitrary degree. Many of thesegroups can be embedded in the automorphism group of a free group.Moreover, the fact that the groups have aspherical presentationsmakes them useful tools in the search to determine the spectrumof exponents for second order Dehn functions. We contributeto this search by giving the first exact calculations of groupswith quadratic and superquadratic exponents. 2000 Mathematical Subject Classification: 20F06, 20F65, 20F69.     

On the Bergman Property for the Automorphism Groups of Relatively Free Groups

A group G is said to have the Bergman property (the propertyof uniformity of finite width) if given any generating X withX = X^–1 of G, we have that G = X^k for some natural k,that is, every element of G is a product of at most k elementsof X. We prove that the automorphism group Aut(N) of any infinitelygenerated free nilpotent group N has the Bergman property. Also,we obtain a partial answer to a question posed by Bergman byestablishing that the automorphism group of a free group ofcountably infinite rank is a group of uniformly finite width.     

An algorithmic construction of group automorphisms and the quantum Yang-Baxter equation

The structure group G of a non-degenerate symmetric set (X,S) is a Bieberbach and a Garside group. We describe a combinatorial method to compute explicitly a group of automorphisms of G and show this group admits a subgroup that preserves the Garside structure. In some special cases, we could also prove the group of automorphisms found is an outer automorphism group.     

Recognizing the Automorphism Groups of Mathieu Groups

It is a well-known fact that characters of a finite group can give important information about the structure of the group. It was also proved by the third author that a finite simple group can be uniquely determined by its character table. Here the authors attempt to investigate how to characterize a finite almost-simple group by using less information of its character table, and successfully characterize the automorphism groups of Mathieu groups by their orders and at most two irreducible character degrees of their character tables.     

On the Residual Finiteness of Outer Automorphisms of Hyperbolic Groups

We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable hyperbolic group is residually finite. As a result, we are able to prove that the group of outer automorphisms of every finitely generated Fuchsian group and of every free-by-finite group is residually finite.     

A Geometric Rational Form for Artin Groups of FC Type

If A is an Artin group whose poset of finite type special subgroups is a flag complex, then A is said to be of FC type. Such groups act cocompactly on a CAT(0) cubical complex with finite type Artin groups as stabilizers. We use the geometry of this complex to obtain a rational normal form for the group.     

The Conjugacy Problem is Solvable in Free-By-Cyclic Groups

We show that the conjugacy problem is solvable in [finitelygenerated free]-by-cyclic groups, by using a result of O. Maslakovathat one can algorithmically find generating sets for the fixedsubgroups of free group automorphisms, and one of P. Brinkmannthat one can determine whether two cyclic words in a free groupare mapped to each other by some power of a given automorphism.We also solve the power conjugacy problem, and give an algorithmto recognize whether two given elements of a finitely generatedfree group are twisted conjugated to each other with respectto a given automorphism. 2000 Mathematics Subject Classification20F10, 20E05.     

The Natural Embedding of Positive Singular Artin Monoids

We explore some combinatorial properties of singular Artin monoids and invoke them to prove that a positive singular Artin monoid of arbitrary Coxeter type necessarily injects into the corresponding singular Artin monoid. This is an extension of L. Pari' result that positive Artin monoids embed in the correpsonding Artin groups: Adjoining inverses of the generators does not produce any new identities between words that do not involve those inverses.     

一类亚循环群的自同构群

本文决定了每 Sylow子群循环的有限群 G的自同构群 ,所得结论包含了徐尚进和 Walls的主要结果 .     

Idempotent-Conjugate Monoids with Nilpotent Unit Groups

Let M be a finite monoid with unit group G such that J-related idempotents in M are conjugate. If G is nilpotent, we prove that the complex monoid algebra CM of M is semisimple if and only if M is an inverse monoid. Conversely let G be a finite group such that for any finite idempotent-conjugate monoid M with unit group G, CM semisimple implies that M is an inverse monoid. We then show that G is a nilpotent group.     

Une démonstration simple de la fidélité de la représentation de Lawrence–Krammer–Paris

The theorem of linearity of the Artin–Tits groups of spherical type and the theorem of injectivity of any Artin–Tits monoid in its group are essentially based on the faithfulness of the Lawrence–Krammer–Paris representation restricted to the monoid. We prove this faithfulness using neither the normal forms of the elements of the monoid nor the closed subsets of the associated root system; only very elementary notions are needed.