No abstract

Representations of the Whitehead manifold W h 3 and Julia sets Annales de la faculté des sciences de Toulouse 6 e série, tome 4, n o 3 (1995), p. 655-694 © Université Paul Sabatier, 1995, tous droits réservés. L'accès aux archives de la revue « Annales de la faculté des sciences de Toulouse » (http://picard.ups-tlse.fr/~annales/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Representations of the Whitehead manifold Wh3 and Julia sets(*) VALENTIN POÉNARU(1) and CORRADO TANASI(2) R,ESUME.-Quand on etend a la variete de Whitehead le theoreme de representation pseudo-spine collapsible pour les spheres d'homotopie, les transversales tendues a l'ensemble des lignes doubles presentent un comportement chaotique bien connu. Ce comportement est engendre par une boucle de feedback dynamique qui engendre aussi des ensembles de Julia. ABSTRACT.-When one extends to the Whitehead manifold the collapsible pseudo-spine representation theorem for homotopy 3-spheres, the tight transversals to the set of double lines present chaotic dynamical behaviour of a very well-known type. This behaviour is generated by a dynamical feedback loop which also generates Julia sets. KEYWORDS : Representations of open simply connected 3-manifolds, Whitehead manifolds, tight transversals to the set of double lines, Chaotic behaviour, Julia sets, Mandelbrot set. .

Abstract. Several authors investigated the properties which are invariant under the passage from a group to its nonabelian tensor square. In the present note we study this problem from the viewpoint of the classes of groups and the methods allow us to prove a result of invariance for some geometric properties of discrete groups.

No abstract

We extend Cannon's notion of k-almost convex groups which requires that for two points x, y on the n-sphere in the Cayley graph which can be joined by a path l 1 of length ~< k, there is a second path 12 in the n-ball, joining x and y, of bounded length ~< N(k). Our k-weakly almost convexity relaxes this condition by requiring only that Ii w 12 bounds a disk of area <<. Cl(k)n 1 -~(k) + C2(k). If M 3 is a closed 3-manifold with 3-weakly almost convex fundamental group, then )z~°J~ 3 = O.Geometria Dedicata 48: 57-81, 1993.