Cohomologie des groupes linéaires,K-théorie de Milnor et groupes de Witt

We construct some remarkable classes in the cohomology of linear groups.     

Nombres de Tamagawa des groupes réductifs quasi-connexes

We define Tamagawa numbers for quasi-connected linear algebraic groups. This category of non-connected linear algebraic groups occurs naturally in the stabilization of the Arthur-Selberg trace formula. An expression for their Tamagawa numbers, needed for the stabilization process, is given in term of cohomological invariants generalizing results of Ono, Sansuc, Kottwitz and Kottwitz & Shestad. Received: 7 March 2000     

Invariance des probabilités de retour sur des groupes localement compacts

We construct an asymptotic invariant for locally compact separable, compactly generated unimodular groups. If G is such a group and if F is a symmetric bounded density on it with second order moment (with respect to a word metric), we show that the asymptotic behavior of $n?F^{1(2n)}(e)$ does not depend on the choice of the density F. As an example we show that the asymptotic of the return probabilities on $\mathrm{sol}(K)$ where K is a p-field behaves like $\exp (?t^{(1/3)})$. In the case where $K={\mathbb{Q}}_p$ this answers a question of Pittet and Saloff-Coste published recently by Mustapha. To cite this article: D. Gretete, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Présentations des groupes de Fischer (I)

This work gives a method for constructing presentation (X, ) for a group generated by a class of 3-transpositions (named here a Fischer class) in which {ie275-1}(G/Z(G)) is simple and non-abelian. The generating set X is contained in D; is the set of relations (xy) ^m(x,y)=1, x and y in X, where m(x, y) is the order of the product xy in G, m(x, y) 3; is a set of relations between elements of X. The group G is constructed as a quotient of the Coxeter group (X, ).These results are applied to symplectic and orthogonal groups over {ie275-2}. Applications to other groups, in particular to the sporadic Fischer groups, will follow later.     

Présentations des groupes de Fischer (II)

In this paper we give Coxeter presentation (X, ) for the three Fischer groupsG=Fi₂₂, Fi₂₃, Fi₂₄; we apply methods exposed in the first part. Each of these groups is generated by a class of 3-transpositions (named here a Fischer class) in which elements ofX are chosen. A subset of is the set of all the relations (xy) ^m(x,y)=1, wherex andy are inX and wherem(x,y) means the order ofxy inG. We obtainG as a specified quotient of the Coxeter group (X, ) with the appropriate diagram .     

Comportement asymptotique des solutions d'une classe d'équations paraboliques et de Hamilton-Jacobi

We study long-time behaviour of the solutions of a class of nonlinear parabolic equations, as well as their Hamilton-Jacobi version. The solutions are shown to converge to either periodic fronts, or to steady solutions, modulo a linear growth in time.     

Une série génératrice des formes paraboliques de Maass

A certain periodic function F_μ,ν(z), an eigenfunction of the Laplacian on the upper half-plane with respect to z depending on some parameters μ, ν, is not automorphic, but the function μ → F_μ,ν(z)−F_μ,ν(−1/0 extends to a larger domain than the function F_μ,ν(z) itself. Consequently, at the poles of this latter function of μ, the coefficients of the polar parts provide non-analytic modular forms: all Maass cusp forms are finite linear combinations of forms obtained in this way, allowing the second parameter ν to vary     

Dégénérescence de sous-groupes discrets des groupes de Lie semi-simples

We prove that a sequence of faithful and discrete (but not necessarily with finite covolume) representations of a finitely generated group Γ in a semisimple Lie group, that goes to infinity, converges to an isometric action of Γ on an affine building, with controlled germ of apartment stabilizers.     

Détermination géométrique de la systole des groupes de triangles

In this Note we give an explicit formula for the length of the shortest geodesic loop for hyperbolic spheres with three singularities of order greater than 3.     

Quelques remarques sur les actions analytiques des réseaux des groupes de Lie de rang supérieur

All the actions considered here are (real) analytic. Let г be a subgroup of finite index of SL(n,). We prove, in particular, the (global) homotopical rigidity, for both its standard affine action on the torus T^a (n ≥ 3), and its standard projective action on the sphere Sⁿ⁻¹ (n, ≥ 4).     

Sur la cohomologie modulo 2 des groupes de Coxeter à trois générateurs

In this Note, we determine all Coxeter groups W of three generators such that the Quillen map: is an isomorphism. C(W) denoting the category whose objects are elementary abelian 2-subgroups of W and whose morphisms are restrictions of conjuguer of W.     

Dynamique des applications holomorphes propres de domaines réguliers et problème de l'injectivité

Summary This paper deals with proper holomorphic self-maps of smoothly bounded pseudoconvex domains in . We study the dynamical properties of their extension to the boundary and show that their non-wandering sets are always contained in the weakly pseudoconvex part of the boundary. In the case of complete circular domains, we combine this fact with an entropy/degree argument to show that the maps are automorphisms. Some of our results remain true in      

Annulation et pureté des groupes de cohomologie rigide associés à des sommes exponentielles

We show a purity property for the rigid cohomology groups associated to exponential sums on a smooth and affine variety. For this purpose, we compare these groups with the p-adic Banach spaces of Dwork's theory constructed by Adolphson and Sperber [1].     

Le problème des diviseurs de zéro pour les groupes de Lie nilpotents

Let G be a non-Abelian, connected, nilpotent Lie group. Then there exist $0\ne \alpha \in C_c(G)$ and $0\ne \xi \in L^2(G)$ such that $\alpha 1\xi =0$, contrary to what happens for the group ${\mathbb{R}}^n$. Moreover, the set of zero divisors is a total subset of $L^2(G)$. This result is first proven for the Heisenberg group $H_n$ where it is based on the existence of non-trivial Schwartz functions f satisfying $f1(X_k+\mathrm{i}{Y}_k)=0$ for $1?k?n$. To cite this article: J. Ludwig et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).