Groupes d'Engel avec la condition maximale sur lesp-sous-groupes finis abéliens

Sans résumé     

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.     

Groupes Réductifs

Sans résumé A certaines périodes durant l’élaboration de ce travail, nous avons bénéficié de l’aide financière de la N.S.F.: le premier auteur à l’Université de Chicago et à Paris (contract GP-2403), le second auteur à l’Institute for Advanced Study (contract GP-779) et à l’Université de Chicago (contract GP-574).     

Groupes pavables et principe variationnel

Sans résumé     

Cocroissance Des Groupes A Petite Simplification

Let be a group presented by e₁,...,e_m|r₁,...,r_k, L the freegroup generated by e₁,...,e_m, and N = Ker(L). Let c_n be thenumber of elements of length n in N. We know that c = lim sup(c_n)^1/n exists and that (2m–1) < c 2m – 1. ifN {1}. We prove that if the group satisfies a condition slightlyweaker than the small cancellation condition C^'() with <1/6, then c(2m–1) when the lengths of the relations r_itend to infinity. A consequence of this result is a theoremof Grigorchuk.     

Groupes fondamentaux motiviques de Tate mixte

We define the category of mixed Tate motives over the ring of S-integers of a number field. We define the motivic fundamental group (made unipotent) of a unirational variety over a number field. We apply this to the study of the motivic fundamental group of the projective line punctured at zero, infinity and all Nth roots of unity.     

Groupes d’automorphismes polynomiaux du plan

We study the embeddings of lattices from simple Lie groups into the group of polynomial automorphisms of the affine plane and answer a question of Dekimpe concerning cristallographic polynomial groups of the plane.      

Groupes algébriques et points rationnels

Résumé. Soit X une variété algébrique définie sur un corps de nombres. Nous comparons diverses obstructions cohomologiques (abéliennes et non-abéliennes) au principe de Hasse et à l'approximation faible sur Xà l'obstruction de Manin.

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Let X be an algebraic variety defined over a number field. We compare miscellaneous (abelian and non-abelian) cohomological obstructions to the Hasse principle and weak approximation for X to the Manin obstruction.

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Received: 14 March 2001 / Revised version: 9 July 2001 / Published online: 28 February 2002     

Groupes vectoriels et schéma de Picard

This Note contains slight variations on a well known theme: linear sub-groups of the Picard functor of a proper scheme over a field k. In particular, we give exemples of a field k, with positive characteristic, and a projective k-scheme X, normal, but not geometrically reduced, such that the neutral component of its Picard functor is representable by a nonzero vectorial group scheme. To cite this article: M. Raynaud, C. R. Acad. Sci. Paris, Ser. I 338 (2004).