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Bachir Bekka

  1. On the spectral theory of groups of affine transformations of compact nilmanifolds.

    Ср, 06/15/2011 - 15:01 — SomNambul
    Authors: Bachir Bekka, Yves Guivarc'h
    Subjects: Dynamical Systems
    Abstract

    Let $N$ be a connected and simply connected nilpotent Lie group, $\Lambda$ a
    lattice in $N$, and $X=N/\Lambda$ the corresponding nilmanifold. Let $Aff(X)$
    be the group of affine transformations of $X$. We characterize the countable
    subgroups $H$ of $Aff(X)$ for which the action of $H$ on $X$ has a spectral
    gap, that is, such that the associated unitary representation $U$ of $H$ on the
    space of functions from $L^2(X)$ with zero mean does not weakly contain the
    trivial representation. Denote by $T$ the maximal torus factor associated to
    $X$.

  2. Groups with faithful irreducible projective unitary representations.

    Чт, 09/23/2010 - 15:02 — SomNambul
    Authors: Bachir Bekka, Pierre de la Harpe
    Subjects: Group Theory
    Abstract

    For a countable group G and a multiplier c on G with values in the circle, we
    study the property of G having a unitary projective c-representation which is
    both irreducible and projectively faithful. We show that this property is
    equivalent to G being the quotient of an appropriate group by its centre. A
    criterion is given in terms of the minisocle of G. Several examples are
    described to show the existence of various behaviours.

  3. Lattices with and lattices without spectral gap.

    Втр, 09/08/2009 - 12:16 — SomNambul
    Authors: Bachir Bekka, Alexander Lubotzky
    Subjects: Dynamical Systems
    Abstract

    The following two results are shown.

    1) Let $G$ be the $k$-rational points of a simple algebraic group over a
    local field $k$ and let $H$ be a lattice in $G.$ Then the regular
    representation of $G$ on $L^2(G/H)$ has a spectral gap (that is, there are
    almost invariant unit vectors in the subspace of functions in $L^2(G/H)$ with
    zero mean).

  4. Lattices with and lattices without spectral gap.

    Втр, 09/08/2009 - 12:16 — SomNambul
    Authors: Bachir Bekka, Alexander Lubotzky
    Subjects: Dynamical Systems
    Abstract

    The following two results are shown.

    1) Let $G$ be the $k$-rational points of a simple algebraic group over a
    local field $k$ and let $H$ be a lattice in $G.$ Then the regular
    representation of $G$ on $L^2(G/H)$ has a spectral gap (that is, there are
    almost invariant unit vectors in the subspace of functions in $L^2(G/H)$ with
    zero mean).

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