We show that each group $\Gamma$ in a class of finitely generated groups introduced in [2] and [3] has Kazhdan’s property (T), and calculate the exact Kazhdan constant of $\Gamma$ with respect to its natural set of generators. These are the first infinite groups shown to have property (T) without making essential use of the theory of representations of linear groups, and the first infinite groups with property (T) for which the exact Kazhdan constant has been calculated. These groups therefore provide answers...

Dans cet article, nous étudions le flot des chambres de Weyl d’une large classe de sous-groupe discrets d’un groupe de Lie semi-simple réel : les groupes de Ping-Pong. Nous montrons que ce flot est mélangeant relativement à la mesure de Patterson-Sullivan ; celle-ci étant infinie en rang $\ge 2$, nous précisons cette propriété de mélange en explicitant sa vitesse dans le direction du vecteur de croissance du groupe.

Soient $X$ une variété algébrique complexe, lisse, irréductible, $E$ et $F$ deux espaces vectoriels complexes de dimension finie et $\mu$ un morphisme de $X$ dans l’espace Lin$(E,F)$ des applications linéaires de $E$ dans $F$. Pour $x\in X$, on note $E\left(x\right)$ et $x·E$ le noyau et l’image de $\mu \left(x\right)$, ${\overline{\mu }}_x$ le morphisme de $X$ dans Lin$\left(E\right(x),F/(x·E\left)\right)$ qui associe à $y$ l’application linéaire $v\mapsto \mu \left(y\right)\left(v\right)+x·E$. Soit i${}_{\mu }$ la dimension minimale de $E\left(x\right)$. On dit que $\mu$ ala propriété $\left(𝐑\right)$ en $x$si i${}_{{\overline{\mu }}_x}$ est inférieur à i${}_{\mu }$. Soient $F^{*}$ le dual de $F$, S$\left(F\right)$ l’algèbre symétrique de $F$, ${ℐ}_{\mu }$ l’idéal de ${𝒪}_X{\otimes }_{ℂ}\mathrm{S}\left(F\right)$ engendré par...

Chevalley’s theorem states that every smooth connected algebraic group over a perfect field is an extension of an abelian variety by a smooth connected affine group. That fails when the base field is not perfect. We define a pseudo-abelian variety over an arbitrary field $k$ to be a smooth connected $k$-group in which every smooth connected affine normal $k$-subgroup is trivial. This gives a new point of view on the classification of algebraic groups: every smooth connected group over a field is an extension...

We give a classification of pseudo-Riemannian weakly symmetric manifolds in dimensions $2$ and $3$, based on the algebraic approach of such spaces through the notion of a pseudo-Riemannian weakly symmetric Lie algebra. We also study the general symmetry of reductive $3$-dimensional pseudo-Riemannian weakly symmetric spaces and particularly prove that a $3$-dimensional reductive $2$-fold symmetric pseudo-Riemannian manifold must be globally symmetric.

For any pair E,F of pseudotopological vector spaces, we endow the space L(E,F) of all continuous linear operators from E into F with a pseudotopology such that, if G is a pseudotopological space, then the mapping L(E,F) × L(F,G) ∋ (f,g) → gf ∈ L(E,G) is continuous. We use this pseudotopology to establish a result about differentiability of certain operator-valued functions related with strongly continuous one-parameter semigroups in Banach spaces, to characterize von Neumann algebras, and to establish...