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

We prove two results on the first $L^p$-cohomology $\overline{H}^{1}_{(p)}(\Gamma)$ of a finitely generated group $\Gamma$: \begin{enumerate} \item [1)] If $N\subset H\subset\Gamma$ is a chain of subgroups, with $N$ non-amenable and normal in $\Gamma$, then $\overline{H}^{1}_{(p)}(\Gamma)=0$ as soon as $\overline{H}^{1}_{(p)}(H)=0$. This allows for a short proof of a result of L\"uck \cite{LucMatAnn}: if $N\lhd\Gamma$, $N$ is infinite, finitely generated as a group, and $\Gamma/N$ contains an element of infinite order, then $\overline{H}^{1}_{(2)}(\Gamma)=0$. \item [2)] If $\Gamma$ acts isometrically, properly discontinuously on a proper $CAT(-1)$ space $X$, with at least 3 limit points in $\partial X$, then for $p$ larger than the critical exponent $e(\Gamma)$ of $\Gamma$ in $X$, one has $\overline{H}^{1}_{(p)}(\Gamma)\neq 0$. 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 $\Gamma$, then $e(G)\leq e(\Gamma)$. \end{enumerate}