# Vanishing and non-vanishing for the first $L_{p}$-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 $L_{p}$-cohomology $H_{(p)}(Γ)$ of a finitely generated group $Γ$: \begin{enumerate} \item [1)] If $N⊂H⊂Γ$ is a chain of subgroups, with $N$ non-amenable and normal in $Γ$, then $H_{(p)}(Γ)=0$ as soon as $H_{(p)}(H)=0$. This allows for a short proof of a result of L\"uck \cite{LucMatAnn}: if $N⊲Γ$, $N$ is infinite, finitely generated as a group, and $Γ/N$ contains an element of infinite order, then $H_{(2)}(Γ)=0$. \item [2)] If $Γ$ acts isometrically, properly discontinuously on a proper $CAT(−1)$ space $X$, with at least 3 limit points in $∂X$, then for $p$ larger than the critical exponent $e(Γ)$ of $Γ$ in $X$, one has $H_{(p)}(Γ)=0$. As a consequence we extend a result of Shalom \cite{Sha}: let $G$ be a cocompact lattice in a rank 1 simple Lie group; if $G$ is isomorphic to $Γ$, then $e(G)≤e(Γ)$. \end{enumerate}

## Cite this article

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

DOI 10.4171/CMH/18