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

Abstract

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}