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

Abstract

We prove two results on the first $L^p$-cohomology ${\overline{H}}_{(p)}^1(\Gamma )$ of a finitely generated group $\Gamma$:

1. If $N\subset H\subset \Gamma$ is a chain of subgroups, with $N$ non-amenable and normal in $\Gamma$, then ${\overline{H}}_{(p)}^1(\Gamma )=0$ as soon as ${\overline{H}}_{(p)}^1(H)=0$. This allows for a short proof of a result of Lück: if $N⊲\Gamma$, $N$ is infinite, finitely generated as a group, and $\Gamma /N$ contains an element of infinite order, then ${\overline{H}}_{(2)}^1(\Gamma )=0$.
2. If $\Gamma$ acts isometrically, properly discontinuously on a proper $\mathrm{CAT}(-1)$ 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 ${\overline{H}}_{(p)}^1(\Gamma )\ne 0$. As a consequence we extend a result of Shalom: let $G$ be a cocompact lattice in a rank 1 simple Lie group; if $G$ is isomorphic to $\Gamma$, then $e(G)\le e(\Gamma )$.