Vanishing and non-vanishing for the first -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


We prove two results on the first -cohomology of a finitely generated group :

  1. If is a chain of subgroups, with non-amenable and normal in , then as soon as . This allows for a short proof of a result of Lück: if , is infinite, finitely generated as a group, and contains an element of infinite order, then .
  2. If acts isometrically, properly discontinuously on a proper space , with at least 3 limit points in , then for larger than the critical exponent of in , one has . As a consequence we extend a result of Shalom: let be a cocompact lattice in a rank 1 simple Lie group; if is isomorphic to , then .

Cite this article

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

DOI 10.4171/CMH/18