Vanishing and non-vanishing for the first LpL^p-cohomology of groups

  • Marc Bourdon

    Université Lille I, Villeneuve d'Ascq, France
  • Florian Martin

    Philip Morris International, Neuchâtel, Switzerland
  • Alain Valette

    Université de Neuchâtel, Switzerland

Abstract

We prove two results on the first LpL^p-cohomology H(p)1(Γ)\overline{H}^{1}_{(p)}(\Gamma) of a finitely generated group Γ\Gamma: \begin{enumerate} \item [1)] If NHΓN\subset H\subset\Gamma is a chain of subgroups, with NN non-amenable and normal in Γ\Gamma, then H(p)1(Γ)=0\overline{H}^{1}_{(p)}(\Gamma)=0 as soon as H(p)1(H)=0\overline{H}^{1}_{(p)}(H)=0. This allows for a short proof of a result of L\"uck \cite{LucMatAnn}: if NΓN\lhd\Gamma, NN is infinite, finitely generated as a group, and Γ/N\Gamma/N contains an element of infinite order, then H(2)1(Γ)=0\overline{H}^{1}_{(2)}(\Gamma)=0. \item [2)] If Γ\Gamma acts isometrically, properly discontinuously on a proper CAT(1)CAT(-1) space XX, with at least 3 limit points in X\partial X, then for pp larger than the critical exponent e(Γ)e(\Gamma) of Γ\Gamma in XX, one has H(p)1(Γ)0\overline{H}^{1}_{(p)}(\Gamma)\neq 0. As a consequence we extend a result of Shalom \cite{Sha}: let GG be a cocompact lattice in a rank 1 simple Lie group; if GG is isomorphic to Γ\Gamma, then e(G)e(Γ)e(G)\leq e(\Gamma). \end{enumerate}

Cite this article

Marc Bourdon, Florian Martin, Alain Valette, Vanishing and non-vanishing for the first LpL^p-cohomology of groups. Comment. Math. Helv. 80 (2005), no. 2, pp. 377–389

DOI 10.4171/CMH/18