Cusps, Kleinian groups and Eisenstein series

Abstract

We study the Eisenstein series associated to the full rank cusps in a complete hyperbolic manifold. We show that given a Kleinian group $\Gamma<\text{Isom}(\mathbb{H}^{n+1})$, each full rank cusp corresponds to a cohomology class in $H^{n}(\Gamma, V)$ where $V$ is either the trivial coefficient or the adjoint representation. Moreover, by computing the intertwining operator, we show that different cusps give rise to linearly independent classes.