Cusps, Kleinian groups and Eisenstein series

TL;DR

This paper investigates Eisenstein series linked to full rank cusps in hyperbolic manifolds, revealing their correspondence with cohomology classes and demonstrating linear independence among different cusps through intertwining operator analysis.

Contribution

It establishes a novel connection between cusps in hyperbolic manifolds and cohomology classes, and analyzes their linear independence via intertwining operators.

Findings

Each full rank cusp corresponds to a cohomology class in H^n(Γ, V).

Different cusps produce linearly independent cohomology classes.

Intertwining operator computation confirms independence of cusp-related classes.

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.