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We prove that for "most" closed 3-dimensional manifolds $M$, the existence of a closed non singular one-form in a given cohomology class $u\in H^1 (M,\bf R)$ is equivalent to the fact that every twisted Alexander polynomial $\Delta^H(M,u) \in {\bf Z}[G/\ker u]$ associated to a normal subgroup with finite index $H < \pi_1(M)$ has a unitary $u$-minimal term., Comment: The statement of the main theorem has been corrected. This version has been accepted for publication in Annales de l'Institut Fourier

We discuss the geometry of some arithmetic orbifolds locally isometric to a product of real hyperbolic spaces of dimension two and three, and prove that certain sequences of non-uniform orbifolds are convergent to this space in a geometric ("Benjamini--Schramm") sense for hyperbolic three--space and a product of hyperbolic planes. We also deal with arbitrary sequences of maximal arithmetic three--dimensional hyperbolic lattices defined over a quadratic or cubic field. A motivating application is the study of Betti numbers of Bianchi groups., Comment: Final version, 51 pages (journal layout). Minor correction to the main theorem from the first version

Let $\H^n$ be the Heisenberg group of topological dimension $2n+1$. We prove that if $n$ is odd, the pair of metric spaces $(\H^n, \H^n)$ does not have the Lipschitz extension property.; Soit H n le groupe d'Heisenberg de dimension topologique 2n+1. On montre que si n est impair, le couple d'espaces métriques (H n , H n) ne possède pas la propriété d'extension lipschitzienne.

We show that the zeroes of the Alexander polynomial of a Lorenz knot all lie in some annulus whose width depends explicitly on the genus and the braid index of the considered knot., 30p., final version, to appear in Ann. Inst. Fourier

We study the ideal triangulation graph T (S) of a punctured surface S of finite type. We show that if S is not the sphere with at most three punctures or the torus with one puncture, then the natural map from the extended mapping class group of S into the simplicial automorphism group of T (S) is an isomorphism. We also show that under the same conditions on S, the graph T (S) equipped with its natural simplicial metric is not Gromov hyperbolic. Thus, from the point of view of Gromov hyperbolicity, the situation of T (S) is different from that of the curve complex of S. AMS Mathematics Subject Classification: Primary 32G15. Secondary 20F38 ; 30F10.

International audience

International audience; On étudie les morphismes d’un groupe infini discret Π dans un groupe de Lie G contenu dans le groupe des difféomorphismes de la droite réelle. À un tel morphisme H, on associe deux ensembles de “bouts” de Π “dans la direction” H. On calcule le nombre de bouts dans plusieurs situations. Dans le cas particulier où Π est de type fini et où G est le groupe des translations, Π n’a qu’un bout dans la direction H si, et seulement si, ils vérifient la propriété de Bieri-Neumann-Strebel.