C*-algebras of Penrose's hyperbolic tilings

Penrose hyperbolic tilings are tilings of the hyperbolic plane which admit, up to affine transformations a finite number of prototiles. In this paper, we give a complete description of the C*-algebras and of the K-theory for such tilings. Since the continuous hull of these tilings have no transversally invariant measure, these C*-algebras are traceless. Nevertheless, harmonic currents give rise to 3-cyclic cocycles and we discuss in this setting a higher-order version of the gap-labelling.Comment: 36 pages. v2: some mistakes corrected, a section on topological invariants of the continuous hull of the Penrose hyperbolic tilings adde

Propriété de décomposition en KK-théorie équivariante pour l'action d'un groupoïde

Appendice de l'article de V. Lafforgue, "K-théorie bivariante pour les algèbres de Banach, groupoïdes et conjecture de Baum-Connes", Journal de l'IMJ 6 (2007), n. 3, pp 415-451.International audienceNous montrons dans cet article que tout élément de KK-théorie équivariante pour l'action d'un groupoïde peut s'écrire comme le produit d'éléments induit par des morphismes et d'inverse en KK-théorie d'éléments inversibles induit par des morphismes. We prove in this paper that every element in equivariant KK-theory with respect to a groupoid action can be decomposed as the product of elements induced in KK-theory by morphisms and of inverse in KK-theory of invertible elements induced by morphisms

Index theory for quasi-crystals I. Computation of the gap-label group

International audienceIn this paper, we give a complete solution to the gap labelling conjecture for quasi-crystals. The method adopted relies on the index theory for laminations, and the main tools are the Connes-Skandalis longitudinal K-theory index morphism together with the measured index formula

An analogue of Serre fibrations for C*-algebra bundles

International audienceWe study an analogue of Serre fibrations in the setting of C-algebra bundles. We derive in this framework a Leray-Serre type spectral sequence. We investigate a class of examples which generalise on one hand principal bundles with a n-torus as structural group and on the other hand non-commutative tori

Homotopy invariance of higher signatures and 3-manifold groups

International audienceFor closed oriented manifolds, we establish oriented homotopy invariance of higher signatures that come from the fundamental group of a large class of orientable 3-manifolds, including the ``piecewise geometric'' ones in the sense of Thurston. In particular, this class, that will be carefully described, is the class of all orientable 3-manifolds if the Thurston Geometrization Conjecture is true. In fact, for this type of groups, we show that the Baum-Connes Conjecture With Coefficients holds. The non-oriented case is also discussed

K-theory for the maximal Roe algebra of certain expanders

We study in this paper the maximal version of the coarse Baum-Connes assembly map for families of expanding graphs arising from residually finite groups. Unlike for the usual Roe algebra, we show that this assembly map is closely related to the (maximal) Baum-Connes assembly map for the group and is an isomorphism for a class of expanders. We also introduce a quantitative Baum-Connes assembly map and discuss its connections to K-theory of (maximal) Roe algebras.Comment: 45 page