Roland Vergnioux

Université de Caen Basse-Normandie, Caen, Lower Normandy, France

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Publications (9)8.07 Total impact

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    Roland Vergnioux
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    ABSTRACT: We study existence, uniqueness and triviality of path cocycles in the quantum Cayley graph of universal discrete quantum groups. In the orthogonal case we find that the unique path cocycle is trivial, in contrast with the case of free groups where it is proper. In the unitary case it is neither bounded nor proper. From this geometrical result we deduce the vanishing of the first L2L2-Betti number of Ao(In)Ao(In).
    Advances in Mathematics 03/2012; 229(5):2686–2711. · 1.37 Impact Factor
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    Roland Vergnioux, Christian Voigt
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    ABSTRACT: In this paper we study the $ K $-theory of free quantum groups in the sense of Wang and Van Daele, more precisely, of free products of free unitary and free orthogonal quantum groups. We show that these quantum groups are $ K $-amenable and establish an analogue of the Pimsner-Voiculescu exact sequence. As a consequence, we obtain in particular an explicit computation of the $ K $-theory of free quantum groups. Our approach relies on a generalization of methods from the Baum-Connes conjecture to the framework of discrete quantum groups. This is based on the categorical reformulation of the Baum-Connes conjecture developed by Meyer and Nest. As a main result we show that free quantum groups have a $ \gamma $-element and that $ \gamma = 1 $. As an important ingredient in the proof we adapt the Dirac-dual Dirac method for groups acting on trees to the quantum case. We use this to extend some permanence properties of the Baum-Connes conjecture to our setting.
    Mathematische Annalen 12/2011; 357(1). · 1.38 Impact Factor
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    Teodor Banica, Roland Vergnioux
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    ABSTRACT: The half-liberated orthogonal group $O_n^*$ appears as intermediate quantum group between the orthogonal group $O_n$, and its free version $O_n^+$. We discuss here its basic algebraic properties, and we classify its irreducible representations. The classification of representations is done by using a certain twisting-type relation between $O_n^*$ and $U_n$, a non abelian discrete group playing the role of weight lattice for $O_n^*$, and a number of methods inspired from the theory of Lie algebras. We use these results for showing that the discrete quantum group dual to $O_n^*$ has polynomial growth.
    Annales- Institut Fourier 02/2009; · 0.53 Impact Factor
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    Teodor Banica, Roland Vergnioux
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    ABSTRACT: We find the fusion rules for the quantum analogues of the complex reflection groups $H_n^s=\mathbb Z_s\wr S_n$. The irreducible representations can be indexed by the elements of the free monoid $\mathbb N^{*s}$, and their tensor products are given by formulae which remind the Clebsch-Gordan rules (which appear at $s=1$). Comment: 33 pages
    Journal of noncommutative geometry 05/2008; · 0.88 Impact Factor
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    ROLAND VERGNIOUX
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    ABSTRACT: We introduce the Property of Rapid Decay for discrete quantum groups by equivalent characterizations that generalize the classical ones. We investigate examples, proving in particular the Property of Rapid Decay for unimodular free quantum groups. We finally check that the applications to the K-theory of the reduced group C -algebras carry over to the quantum case.
    JOURNAL OF OPERATOR THEORY 01/2007; 2(2). · 0.57 Impact Factor
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    Teodor Banica, Roland Vergnioux
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    ABSTRACT: We discuss the notion of growth for discrete quantum groups, with a number of general considerations, and with some explicit computations. Of particular interest is the quantum analogue of Gromov's estimate regarding polynomial growth: we formulate the precise question, and we verify it for the duals of classical Lie groups. Comment: 17 pages
    Infinite Dimensional Analysis Quantum Probability and Related Topics 11/2006; · 0.39 Impact Factor
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    Stefaan Vaes, Roland Vergnioux
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    ABSTRACT: We study the C*-algebras and von Neumann algebras associated with the universal discrete quantum groups. They give rise to full prime factors and simple exact C*-algebras. The main tool in our work is the study of an amenable boundary action, yielding the Akemann-Ostrand property. Finally, this boundary can be identified with the Martin or the Poisson boundary of a quantum random walk.
    Duke Mathematical Journal 10/2005; · 1.70 Impact Factor
  • Roland Vergnioux
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    ABSTRACT: The Cayley graph of a discrete group is an important object in geometric group theory and noncommutative geometry. The main goal of this article is to define a generalisation of a Cayley graph for discrete quantum groups in the sense of Woronowicz. At least two approaches are put forward. The first is a classical graph which for discrete groups coincides with the usual Cayley graph. The second is a hilbertian Cayley graph, an object of a more noncommutative character. Quantum Cayley trees, a special class of quantum Cayley graphs, are introduced and studied more intensively including their geometry at infinity. The theory is illustrated by several examples and explicit calculations involving free quantum groups. As an application, a version of the factorisation property established for free groups by Akemann and Ostrand is shown for free quantum groups and applications in KK-theory (definition of a Julg-Valette element) are given.
    Journal Fur Die Reine Und Angewandte Mathematik - J REINE ANGEW MATH. 01/2005; 2005(580):101-138.
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    Roland Vergnioux
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    ABSTRACT: The basic notions and results of equivariant KK-theory concerning crossed products can be extended to the case of locally compact quantum groups. We recall these constructions and prove some useful properties of subgroups and amalgamated free products of discrete quantum groups. Using these properties and a quantum analogue of the Bass-Serre tree, we establish the K-amenability of amalgamated free products of amenable discrete quantum groups.
    Journal of Functional Analysis 01/2004; · 1.25 Impact Factor